Polynomial Degree and Finite Differences


 Dylan Greene
 1 years ago
 Views:
Transcription
1 CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson, you Learn the terminology associated with polynomials Use the finite differences method to determine the degree of a polynomial Find a polynomial function that models a set of data A polynomial in one variale is any epression that can e written in the form a n n a n1 n1 a 1 1 a 0 where is a variale, the eponents are nonnegative integers, and the coefficients are real numers. A function in the form f() a n n a n1 n1 a 1 1 a 0 is a polynomial function. The degree of a polynomial or polynomial function is the power of the term with the greatest eponent. If the degrees of the terms of a polynomial decrease from left to right, the polynomial is in general form. The polynomials elow are in general form. 1st degree nd degree 3rd degree th degree A polynomial with one term, such as 5, is called a monomial. A polynomial with two terms, such as 3 7, is called a inomial. A polynomial with three terms, such as 1.8, is called a trinomial. Polynomials with more than three terms, such as , are usually just called polynomials. For linear functions, when the values are evenly spaced, the differences in the corresponding yvalues are constant. This is not true for polynomial functions of higher degree. However, for nddegree polynomials, the differences of the differences, called the second differences and areviated D,are constant. For 3rddegree polynomials, the differences of the second differences, called the third differences and areviated D 3,are constant. This is illustrated in the tales on page 361 of your ook. If you have a set of data with equally spaced values, you can find the degree of the polynomial function that fits the data (if there is a polynomial function that fits the data) y analyzing the differences in yvalues. This technique, called the finite differences method, is illustrated in the eample in your ook. Read that eample carefully. Notice that the finite differences method determines only the degree of the polynomial. To find the eact equation for the polynomial function, you need to find the coefficients y solving a system of equations or using some other method. In the eample, the D values are equal. When you use eperimental data, you may have to settle for differences that are nearly equal. Investigation: Free Fall If you have a motion sensor, collect the (time, height) data as descried in Step 1 in your ook. If not, use these sample data. (The values in the last two columns are calculated in Step.) Discovering Advanced Algera Condensed Lessons CHAPTER 7 93
2 Lesson 7.1 Polynomial Degree and Finite Differences Complete Steps 6 in your ook. The results given are ased on the sample data. Step The first and second differences, D 1 and D,are shown in the tale at right. For these data, we can stop with the second differences ecause they are nearly constant. Step 3 The three plots are shown elow (L1, L) (L3, L) (L5, L6) Time (s) Height (m) y D 1 D [0, 1, 0.5, 0,.5, 0.5] [0, 1, 0.5, 0.5, 0.5, 0.5] [0, 1, 0.5, 0.5, 0.5, 0.5] Step The graph of (L1, L) appears paraolic, suggesting that the correct model may e a nddegree polynomial function. The graph of (L3, L) shows that the first differences are not constant ecause they decrease in a linear fashion. The graph of (L5, L6) shows that the second differences are nearly constant, so the correct model should e a nddegree polynomial function. Step 5 A nddegree polynomial in the form y a c fits the data. Step 6 To write the system, choose three data points. For each point, write an equation y sustituting the time and height values for and y in the equation y a c. The following system is ased on the values (0, ), (0., 1.80), and (0., 1.16). c a 0. c a 0. c 1.16 One way to solve this system is y writing the matri equation a c 1.16 and solving using an inverse matri. The solution is a.9, 0, and c, so the equation that fits the data is y.9. Read the remainder of the lesson in your ook. 9 CHAPTER 7 Discovering Advanced Algera Condensed Lessons
3 CONDENSED LESSON 7. Equivalent Quadratic Forms In this lesson, you Learn aout the verte form and factored form of a quadratic equation and the information each form reveals aout the graph Use the zeroproduct property to find the roots of a factored equation Write a quadratic model for a data set in verte, general, and factored form A nddegree polynomial function is called a quadratic function. In Lesson 7.1, you learned that the general form of a quadratic function is y a c. In this lesson you will eplore other forms of a quadratic function. You know that every quadratic function is a transformation of y.when a quadratic function is written in the form y k a h or y a h k, you can tell that the verte of the paraola is (h, k) and that the vertical and horizontal scale factors are a and. Conversely, if you know the verte of a paraola and you know (or can find) the scale factors, you can write its equation in one of these forms. The quadratic function y a h k can e rewritten in the form a y ( h) k. The coefficient a comines the two scale factors into one vertical scale factor. In the verte form of a quadratic equation, y a( h) k, this single scale factor is simply denoted a. From this form, you can identify the verte, (h, k), and the vertical scale factor, a. If you know the verte of a paraola and the vertical scale factor, you can write an equation in verte form. The zeroproduct property states that for all real numers a and, if a 0, then a 0, or 0, or a 0 and 0. For eample, if 3( 7) 0, then 3 0 or 7 0. Therefore, 0 or 7. The solutions to an equation in the form f() 0 are called the roots of the equation, so 0 and 7 are the roots of 3( 7) 0. The intercepts of a function are also called the zeros of the function (ecause the yvalues are 0). The function y 1.( 5.6)( 3.1), given in Eample A in your ook, is said to e in factored form ecause it is written as the product of factors. The zeros of the function are the solutions of the equation 1.( 5.6)( 3.1) 0. Eample A shows how you can use the zeroproduct property to find the zeros of the function. In general, the factored form of a quadratic function is y a r 1 r. From this form, you can identify the intercepts (or zeros), r 1 and r, and the vertical scale factor, a. Conversely, if you know the intercepts of a paraola and know (or can find) the vertical scale factor, then you can write the equation in factored form. Read Eample B carefully. Investigation: Rolling Along Read the Procedure Note and Steps 1 3 in your ook. Make sure you can visualize how the eperiment works. Use these sample data to complete Steps 8, and then compare your results to those elow. (These data have een adjusted for the position of the starting line as descried in Step 3.) Discovering Advanced Algera Condensed Lessons CHAPTER 7 95
4 Lesson 7. Equivalent Quadratic Forms Time (s) Distance from line (m), y Time (s) Distance from line (m), y Time (s) Distance from line (m), y Step At right is a graph of the data. The data have a paraolic shape, so they can e modeled with a quadratic function. Ignoring the first and last few data points (when the can started and stopped), the second differences, D,are almost constant, at around 0.06, which implies that a quadratic model is appropriate. Step 5 The coordinates of the verte are (3.,.57). Consider (5., 0.897) to e the image of (1, 1). The horizontal and vertical distances of (1, 1) from the verte of y are oth 1. The horizontal distance of (5., 0.897) from the verte, (3.,.57), is, and the vertical distance is So, the horizontal and vertical scale factors are and 3.36, respectively. This can e represented as the single vertical scale factor Therefore, the verte form of a model for the data is y 0.8( 3.).57. Step 6 Sustituting the points (1, 0.56), (3,.56), and (5, 1.93) into the general form, y a c, gives the system a c a 3 c.56 5a 5 c 1.93 The solution to this system is a 0.81, 5.10, and c 3.7, so the general form of the equation is y Step 7 The intercepts are aout (0.9, 0) and (5.5, 0). The scale factor, found in Step 5, is 0.8. So, the intercept form of the equation is y 0.8( 0.9)( 5.5). [1, 7, 1, 1, 7, 1] Step 8 In general, you use the verte form when you know either the verte and the scale factor or the verte and one other point you can use to find the scale factor. You use the general form when you know any three points. You use the factored form when you know the intercepts and at least one other point you can use to find the scale factor. [1, 7, 1, 1, 7, 1] 96 CHAPTER 7 Discovering Advanced Algera Condensed Lessons
5 CONDENSED LESSON 7.3 Completing the Square In this lesson, you Use the method of completing the square to find the verte of a paraola whose equation is given in general form Solve prolems involving projectile motion Many realworld prolems involve finding the minimum or maimum value of a function. For quadratic functions, the maimum or minimum value occurs at the verte. If you are given an equation in verte form, you can easily find the coordinates of a paraola s verte. It is also fairly straightforward to find the verte if the equation is in factored form. It gets more complicated if the equation is in general form. In this lesson you will learn a technique for converting a quadratic equation from general form to verte form. Projectile motion the rising or falling of ojects under the influence of gravity can e modeled y quadratic functions. The height of a projectile depends on the height from which it is thrown, the upward velocity with which it is thrown, and the effect of gravity pulling down on the oject. The height can e modeled y the function y 1 g v 0 s 0 where is the time in seconds, y is the height (in m or ft), g is the acceleration due to gravity (either 9.8 m/s or 3 ft/s ), v 0 is the initial upward velocity of the oject (in either m/s or ft/s), and s 0 is the initial height of the oject (in m or ft). Read Eample A in your ook. It illustrates how to write a projectile motion equation when you know only the intercepts (the two times when the height is 0) and how to use the intercepts to find the coordinates of the verte. It is important to understand that the coordinate of a paraola s verte is the mean of the intercepts. This fact allows you to find the verte of a paraola from the factored form of its equation. Look at the diagrams for ( 5) and (a ) in your ook. Make sure you understand the pattern descried after the diagrams. Investigation: Complete the Square Complete the investigation in your ook. When you are finished, compare your answers to those elow. Step 1 a. You must add ( 3) 9 c. Enter 6 as Y1 and ( 3) 9 as Y, and verify that the tale values or the graphs are the same for oth epressions. Step a ( 3) 13 c. Enter 6 as Y1 and ( 3) 13 as Y, and verify that the tale values or the graphs are the same for oth epressions. Discovering Advanced Algera Condensed Lessons CHAPTER 7 97
6 Lesson 7.3 Completing the Square Step 3 a. Focus on 1. To complete a perfect square, you need to add 9. (This gives an epression in the form a a,where a and 7.) You need to sutract 9 to compensate. So, ( 7) 6. To make a perfect square, you must add,or. You need to sutract to compensate. So, Step a. 6 1 ( 3) 1 Factor 6. Step Complete the square. You add 9, so you must sutract Write in the form a( h) k.. a 10 7 a 1 a0 7 Factor a 10. a c a a c Step 6 y a a h a a 1 0 a a5 a a 5 7 Complete the square. You add a 5 a, so you must sutract a 5 a. a 5 a 7 a5 Write in the form a( h) k. a a a a c a a c a Using the results from Step 5, you can write y a c as c. a This is in verte form, y a( h) k, where and k c. So, the verte is, c. a a a Based on your work in the investigation, you now know two ways to find the verte, (h, k), of a quadratic function given in general form, y a c. 1. You can use the process of completing the square to rewrite the equation in verte form, y a( h) k, and then read the verte from the rewritten equation.. You can use the formulas h a and k = c of the verte directly. a to calculate the coordinates You can use either method, ut make sure you are comfortale with completing the square ecause it will come up in your later work. Eample B in your ook demonstrates oth methods. Work through that eample carefully. Eample C applies what you learned in the investigation to solve a projectile motion prolem. Try to solve the prolem on your own efore reading the solution. 98 CHAPTER 7 Discovering Advanced Algera Condensed Lessons
7 CONDENSED LESSON 7. The Quadratic Formula In this lesson, you Learn how the quadratic formula is derived Use the quadratic formula to solve projectile motion prolems You can use a graph to approimate the intercepts of a quadratic function. If you can write the equation of the function in factored form, you can find the eact values of its intercepts. Unfortunately, most quadratic equations cannot easily e converted to factored form. In this lesson you will learn a method that will allow you to find the eact intercepts of any quadratic function. Read Eample A in your ook carefully, and then read the eample elow. EXAMPLE Find the intercepts of y 7 1. Solution The intercepts are the solutions of See if you can supply the reason for each step in the solution elow. Note that the first four steps involve rewriting the left side in the form a( h) k The intercepts are and = The series of equations after Eample A in your ook shows how you can derive the quadratic formula y following the same steps used in Eample A. The quadratic formula, ac a gives the general solution to a quadratic equation in the form a c 0. Follow along with the steps in the derivation, using a pencil and paper. To make sure you understand the quadratic formula, use it to verify that the solutions of are and. Discovering Advanced Algera Condensed Lessons CHAPTER 7 99
8 Lesson 7. The Quadratic Formula Investigation: How High Can You Go? Complete the investigation in your ook, and then compare your answers to those elow. Step 1 The equation is y , where y is the height and is the time in seconds. (If you answered this question incorrectly, look ack at the discussion of projectile motion in Lesson 7.3.) Step The equation is Step 3 In a c 0 form, the equation is For this equation, a 16, 88, and c 1. Sustituting these values into the quadratic formula gives (16 )(1 ) (16) 3 3 So, or So, the all is feet aove the ground 0.5 second after it is hit (on the way up) and 5.5 seconds after it is hit (on the way down). Step The verte is the maimum. The all reaches the maimum height only once. The all reaches other heights once on the way up and once on the way down, ut the maimum point is the height where the all changes directions, so it is reached only once. Step 5 The equation is In a c 0 form, the equation is For this equation, a 16, 88, and c 11. Sustituting these values into the quadratic formula gives (16 )(1 1) (16) The all reaches a maimum height.75 seconds after it is hit. The fact that there is only one solution ecomes apparent when you realize that the value under the square root sign is 0. Step 6 The equation is In a c 0 form, the equation is For this equation, a 16, 88, and c 197. Sustituting these values into the quadratic formula gives ( )(19 7) (16) 3 The value under the square root sign is negative. Because the square root of a negative numer is not a real numer, the equation has no realnumer solution. Your work in the investigation shows that when the value under the square root sign, ac, is 0, the equation a c 0 has only one solution, and when the value under the square root sign, ac, is negative, the equation a c 0 has no realnumer solutions. This means that if you are given a quadratic equation in the general form, you can use the value of ac to determine whether the graph will have zero, one, or two intercepts. Eample B in your ook shows the importance of writing an equation in general form efore you attempt to apply the quadratic formula. Read the eample carefully. 100 CHAPTER 7 Discovering Advanced Algera Condensed Lessons
9 CONDENSED LESSON 7.5 Comple Numers In this lesson, you Learn that some polynomial equations have solutions that are comple numers Learn how to add, sutract, multiply, and divide comple numers The graph of y.5 has no intercepts. y If you use the quadratic formula to attempt to find the intercepts, you get 1 1(1)( ) (1) The numers and are not real numers ecause they involve the square root of a negative numer. Numers that include the real numers as well as the square roots of negative numers are called comple numers. Defining the set of comple numers makes it possile to solve equations such as.5 0 and 0, which have no solutions in the set of real numers. The square roots of negative numers are epressed using an imaginary unit called i, defined y i 1 or i 1. You can rewrite 9 as 9 1, or 3i. Therefore, the two solutions to the quadratic equation aove can e written as 1 3i and 1 3i, or 1 3 i and 1 3 i. These two solutions are a conjugate pair, meaning that one is in the form a + i and the other is in the form a i. The two numers in a comple pair are comple conjugates. Roots of polynomial equations can e real numers or nonreal comple numers, or there may e some of each. However, as long as the polynomial has realnumer coefficients, any nonreal roots will come in conjugate pairs, such as 3i and 3i or 6 5i and 6 5i. Your ook defines a comple numer as a numer in the form a i, where a and are real numers and i 1. The numer a is called the real part, and the numer is called the imaginary part. The set of comple numers includes all real numers and all imaginary numers. Look at the diagram on page 39 of your ook, which shows the relationship etween these numers, and some other sets of numers you may e familiar with, as well as eamples of numers in each set. Then read the eample in your ook, which shows how to solve the equation 3 0. Discovering Advanced Algera Condensed Lessons CHAPTER 7 101
10 Lesson 7.5 Comple Numers Investigation: Comple Arithmetic In this investigation you discover the rules for computing with comple numers. Part 1: Addition and Sutraction Adding and sutracting comple numers is similar to comining like terms. Use your calculator to add or sutract the numers in Part 1a d in your ook. Then, make a conjecture aout how to add comple numers without a calculator. Below are the solutions and a possile conjecture. a. 5 i. 5 3i c. 1 9i d. 3 i Possile conjecture: To add two comple numers, add the real parts and add the imaginary parts. In symols, (a i) (c di) (a c) ( d)i. Part : Multiplication Multiplying the comple numers a i and c di is very similar to multiplying the inomials a and c d. You just need to keep in mind that i 1. Multiply the comple numers in Part a d, and epress the answers in the form a i. The answers are elow. a. ( i)(3 5i) 3 5i i 3 i 5i Epand as you would for a product of inomials. 6 10i 1i 0i 6 i 0i 6 i 0(1) Multiply within each term. Comine 10i and 1i. i 1 6 i Comine 6 and i c. 1 16i d. 8 16i Part 3: The Comple Conjugates Complete Part 3a d, which involves finding either the sum or product of a comple numer and its conjugate. The answers are given elow. a.. 1 c. 0 d. 3 Possile generalizations: The sum of a numer and its conjugate is a real numer: (a i) (a i) a. The product of a real numer and its conjugate is a real numer: (a i)(a i) a. 10 CHAPTER 7 Discovering Advanced Algera Condensed Lessons
11 Lesson 7.5 Comple Numers Part : Division To divide two comple numers, write the division prolem as a fraction, conjugate of denominator multiply y conjugate of denominator (to change the denominator to a real numer), and then write the result in the form a i. Divide the numers in Part a d. Here are the answers. a. 7 i 1 i 7 i 1 i 1 i 1 conjugate of denominator Multiply y. i conjugate of denominator 5 9i.5.5i Multiply. The denominator ecomes a real numer. Divide i c i d i Comple numers can e graphed on a comple plane, where the horizontal ais is the real ais and the vertical ais is the imaginary ais. The numer a i is represented y the point with coordinates (a, ). The numers 3 i and i are graphed elow. 5 i 5 Imaginary ais 3 i 5 Real ais 5 Discovering Advanced Algera Condensed Lessons CHAPTER 7 103
12 CONDENSED LESSON 7.6 Factoring Polynomials In this lesson, you Learn aout cuic functions Use the intercepts of a polynomial function to help you write the function in factored form The polynomial equations y 6 9 and y ( 3)( 3) are equivalent. The first is in general form, and the second is in factored form. Writing a polynomial equation in factored form is useful for finding the intercepts, or zeros, of the function. In this lesson you will learn some techniques for writing higherdegree polynomials in factored form. A 3rddegree polynomial function is called a cuic function. At right is a graph of the cuic function y The intercepts of the function are, 1.5, and 1.5, so its factored equation must e in the form y a( )( 1.5)( 1.5). To find the value of a, you can sustitute the coordinates of another point on the curve. The yintercept is (0, 36). Sustituting this point into the equation gives 36 a()(1.5)(1.5). So, a, and the factored form of the equation is y ( )( 1.5)( 1.5) (, 0) 5 y 50 (0, 36) ( 1.5, 0) 50 (1.5, 0) 5 Investigation: The Bo Factory You can make a o from a 16y0unit sheet of paper, y cutting squares of side length from the corners and folding the sides up. Follow the Procedure Note in your ook to construct oes for several different integer values of. Record the dimensions and volume of each o. (If you don t want to construct the oes, try topicture them in your mind.) Complete the investigation, and then compare your results to those elow. Step 1 Here are the results for integer values from 1 to Volume Length Width Height y Step The dimensions of the oes are 0, 16, and. Therefore, the volume function is y (16 )(0 )(). Discovering Advanced Algera Condensed Lessons CHAPTER 7 105
13 Lesson 7.6 Factoring Polynomials Step 3 The data points lie on the graph of the function. [, 1, 1, 00, 500, 100] Step If you were to epand (16 )(0 )(), the result would e a polynomial, and the highest power of would e 3. Therefore, the function is a 3rddegree polynomial function. Step 5 The intercepts of the graph are 0, 8, and 10, so the function is y ( 8)( 10). Step 6 The graphs have the same intercepts and general shape ut different vertical scale factors. A vertical scale factor of makes them equivalent: y ( 8)( 10). Step 7 If 0, there are no sides to fold up, so a o cannot e formed. For 8, 8unitwide strips would e cut off the sides of the sheet. Folding up the sides would mean folding the remaining strip in half, which would not form a o. 8 8 [, 1, 1, 00, 500, 100] 8 Cut off Cut off Cut off Cut off A value of 10 is impossile ecause it is more than half the length of the shorter side of the sheet. Only a domain of 0 8 makes sense in this situation. By zooming and tracing to find the coordinates of the high point of the graph, you can find that the value of aout.9 maimizes the volume. Work through the eample in your ook, which asks you to find the factored form of a polynomial function y using the intercepts of the graph. This method works well when the zeros of a function are integer values. Unfortunately, this is not always the case. Sometimes the zeros of a polynomial are not nice rational or integer values, and sometimes they are not even real numers. With quadratic functions, if you cannot find the zeros y factoring or making a graph, you can always use the quadratic formula. Once you know the zeros, r 1 and r,you can write the polynomial in the form y a r 1 r.read the remainder of the lesson in your ook, and then read the eample on the net page. 106 CHAPTER 7 Discovering Advanced Algera Condensed Lessons
14 Lesson 7.6 Factoring Polynomials EXAMPLE Write the equation of the function elow in factored form. y 6 (, ) Solution The factored equation is in the form y a r 1 r,where r 1 and r are the zeros. From the graph, you can see that the only realnumer zero is 3. If the other zero were a nonreal numer, then its conjugate would also e a zero. This would mean there are three zeros, which is not possile. So, 3 must e a doule zero. This means that the function is in the form y a( 3)( 3), or y a( 3).To find the value of a, sustitute (, ): a(1),so a. The factored form of the function is y ( 3). Discovering Advanced Algera Condensed Lessons CHAPTER 7 107
15 CONDENSED LESSON 7.7 HigherDegree Polynomials In this lesson, you Descrie the etreme values and end ehavior of polynomial functions Solve a prolem that involves maimizing a polynomial function Write equations for polynomial functions with given intercepts Polynomials with degree 3 or higher are often referred to as higherdegree polynomials. At right is the graph of the polynomial y ( 3),or y The zeroproduct property tells you that the zeros are 0 and 3 the values of for which y 0. The intercepts of the graph confirm this. The graph has other key features in addition to the intercepts. For eample, the point (1, ) is called a local minimum ecause it is lower than the other points near it. The point (3, 0) is called a local maimum ecause it is higher than the other points near it. You can also descrie the end ehavior of the graph that is, what happens to the graph as increases in the positive and negative directions. For this graph, as increases in the positive direction, y increases in the negative direction. As increases in the negative direction, y increases in the positive direction. The introduction to Lesson 7.7 in your ook gives another eample of a 3rddegree polynomial and its graph. The graph of a polynomial function with real coefficients has a yintercept, possily one or more intercepts, and other features such as local maimums or minimums and end ehavior. The maimums and minimums are called etreme values. y (1, ) 6 6 Investigation: The Largest Triangle Start with an 8.5y11 in. sheet of paper. In centimeters, the dimensions are 1.5 y 8 cm. Orient the paper so that the long side is horizontal. Fold the upper left corner so that it touches some point on the ottom edge. Find the area, in cm,ofthe triangle formed in the lower left corner. A A What distance,, along the ottom of the paper produces the triangle with greatest area? To answer this question, first find a formula for the area of the triangle in terms of. Try to do this on your own efore reading on. Here is one way to find the formula: Let h e the height of the triangle. Then the hypotenuse has length 1.5 h. (Why?) h 1.5h Discovering Advanced Algera Condensed Lessons CHAPTER 7 109
16 Lesson 7.7 HigherDegree Polynomials Use the Pythagorean Theorem to help you write h in terms of. h (1.5 h), so h Now, you can write a formula for the area, y. y At right is a graph of the area function. If you trace the graph, you ll find that the maimum point is aout (1.,.5). Therefore, the value of that gives the greatest area is aout 1. cm. The maimum area is aout.5 cm. [5, 5, 5, 10, 50, 10] Eample A in your ook shows you how to find the equation for a polynomial with given intercepts and yintercept. Read this eample carefully. To test your understanding, find a polynomial function with intercepts 6,, and 1, and yintercept 60. (One answer is y 5( 6)( )( 1).) Graphs A D on page 07 of your ook show the possile shapes for the graph of a 3rddegree polynomial function. Graph A is the graph of the parent function y 3.The graph of any other 3rddegree polynomial will e a transformation of Graph A. Eample B in your ook shows you how to find a polynomial function with given zeros when some of the zeros are comple. The key to finding the solution is to recall that comple zeros come in conjugate pairs. Read that eample carefully, and then read the eample elow. EXAMPLE Solution Find a thdegree polynomial function with real coefficients and zeros, 3, and 1 i. Comple zeros occur in conjugate pairs, so 1 i must also e a zero. So, one possile function, in factored form, is y ( )( 3)( (1 i))( (1 i)) Multiply the factors to get a polynomial in general form. y ( )( 3)( (1 i))( (1 i)) ( 6)( (1 i) (1 i) (1 i)(1 i)) ( 6)( i i ) ( 6)( ) Check the solution y making a graph. (You will see only the real zeros.) Note that an nthdegree polynomial function always has n zeros. However, some of the zeros may e nonreal numers, the function may not have n intercepts. 110 CHAPTER 7 Discovering Advanced Algera Condensed Lessons
17 CONDENSED LESSON 7.8 More Aout Finding Solutions In this lesson, you Use long division to find the roots of a higherdegree polynomial Use the Rational Root Theorem to find all the possile rational roots of a polynomial Use synthetic division to divide a polynomial y a linear factor You can find the zeros of a quadratic function y factoring or y using the quadratic formula. You can sometimes use a graph to find the zeros of higherdegree polynomials, ut this method may give only an approimation of real zeros and won t work at all to find nonreal zeros. In this lesson you will learn a method for finding the eact zeros, oth real and nonreal, of many higherdegree polynomials. Eample A in your ook shows that if you know some of the zeros of a polynomial function, you can sometimes use long division to find the other roots. Follow along with this eample, using a pencil and paper. Make sure you understand each step. To confirm that a value is a zero of a polynomial function, you sustitute it into the equation to confirm that the function value is zero. This process uses the Factor Theorem, which states that ( r) is a factor of a polynomial function P() if and only if P(r) 0. When you divide polynomials, e sure to write oth the divisor and the dividend so that the terms are in order of decreasing degree. If a degree is missing, insert a term with coefficient 0 as a placeholder. For eample, to divide y 9, rewrite as and rewrite 9 as 0 9. The division prolem elow shows that In Eample A, you found some of the zeros y looking at the graph. If the intercepts of a graph are not integers, identifying the zeros can e difficult. The Rational Root Theorem tells you which rational numers might e zeros. It states that if the polynomial equation P() 0 has rational roots, then they are of the form p,where q p is a factor of the constant term and q is a factor of the leading coefficient. Note that this theorem helps you find only rational roots. Eample B shows how the theorem is used. Work through that eample, and then read the eample on the net page. Discovering Advanced Algera Condensed Lessons CHAPTER 7 111
18 Lesson 7.8 More Aout Finding Solutions EXAMPLE Find the roots of Solution The graph of the function y = appears at right. None of the intercepts are integers. The Rational Root Theorem tells you that any rational root will e a factor of divided y a factor of 7. The factors of are 1,, 3,, 6, 8, 1, and. The factors of 7 are 1 and 7. You know there are no integer roots, so you need to consider only 1 7, 7, 3 7, 7, 6 7, 8 7, 1 7, and 7. The graph indicates that one of the roots is etween 3 and. None of these possiilities are in that interval. Another root is a little less than 1.This could e 3 7.Try sustituting 3 7 into the polynomial [3, 3, 1, 60, 60, 10] So, 3 7 is a root, which means 3 7 is a factor. Use division to divide out this factor So, is equivalent to To find the other roots, solve The solutions are 8. So, the roots are 3, 7 8, and 8. Synthetic division is a shortcut method for dividing a polynomial y a linear factor. Read the remainder of the lesson in your ook to see how to use synthetic 7 division. Below is an eample using synthetic division to find Note that in the eample aove you found this same quotient using long division. Known zero 3_ 7 The result shows that Coefficients of Add Add Add Bring down 3_ _ _ , so CHAPTER 7 Discovering Advanced Algera Condensed Lessons
Completing the Square
3.5 Completing the Square Essential Question How can you complete the square for a quadratic epression? Using Algera Tiles to Complete the Square Work with a partner. Use algera tiles to complete the square
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,
More information9.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED LESSON
CONDENSED LESSON 9.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations solve
More informationMathematics Background
UNIT OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND UNIT INTRODUCTION Patterns of Change and Relationships The introduction to this Unit points out to students that throughout their study of Connected
More informationReview: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a
Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6. The student will simplify polynomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a
More informationFinding Complex Solutions of Quadratic Equations
y  y    x x Locker LESSON.3 Finding Complex Solutions of Quadratic Equations Texas Math Standards The student is expected to: A..F Solve quadratic and square root equations. Mathematical Processes
More informationCopyrighted by Gabriel Tang B.Ed., B.Sc. Page 111.
Algera Chapter : Polnomial and Rational Functions Chapter : Polnomial and Rational Functions  Polnomial Functions and Their Graphs Polnomial Functions:  a function that consists of a polnomial epression
More informationMath Analysis Chapter 2 Notes: Polynomial and Rational Functions
Math Analysis Chapter Notes: Polynomial and Rational Functions Day 13: Section 1 Comple Numbers; Sections  Quadratic Functions 1: Comple Numbers After completing section 1 you should be able to do
More informationSolving Systems of Linear Equations Symbolically
" Solving Systems of Linear Equations Symolically Every day of the year, thousands of airline flights crisscross the United States to connect large and small cities. Each flight follows a plan filed with
More information3.5 Solving Quadratic Equations by the
www.ck1.org Chapter 3. Quadratic Equations and Quadratic Functions 3.5 Solving Quadratic Equations y the Quadratic Formula Learning ojectives Solve quadratic equations using the quadratic formula. Identify
More informationAlgebra I Notes Unit Nine: Exponential Expressions
Algera I Notes Unit Nine: Eponential Epressions Syllaus Ojectives: 7. The student will determine the value of eponential epressions using a variety of methods. 7. The student will simplify algeraic epressions
More informationAdditional Factoring Examples:
Honors Algebra 3 Solving Quadratic Equations by Graphing and Factoring Learning Targets 1. I can solve quadratic equations by graphing. I can solve quadratic equations by factoring 3. I can write a quadratic
More informationPolynomials and Polynomial Functions
Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial
More informationGraphs and polynomials
5_6_56_MQVMM  _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Graphs and polnomials VCEcoverage Areas of stud Units & Functions and graphs Algera In this chapter A The inomial
More informationAlgebra I Notes Concept 00b: Review Properties of Integer Exponents
Algera I Notes Concept 00: Review Properties of Integer Eponents In Algera I, a review of properties of integer eponents may e required. Students egin their eploration of power under the Common Core in
More informationObjectives To solve equations by completing the square To rewrite functions by completing the square
46 Completing the Square Content Standard Reviews A.REI.4. Solve quadratic equations y... completing the square... Ojectives To solve equations y completing the square To rewrite functions y completing
More informationSection 2.1: Reduce Rational Expressions
CHAPTER Section.: Reduce Rational Expressions Section.: Reduce Rational Expressions Ojective: Reduce rational expressions y dividing out common factors. A rational expression is a quotient of polynomials.
More informationSolving and Graphing Polynomials
UNIT 9 Solving and Graphing Polynomials You can see laminar and turbulent fl ow in a fountain. Copyright 009, K1 Inc. All rights reserved. This material may not be reproduced in whole or in part, including
More informationModule 2, Section 2 Solving Equations
Principles of Mathematics Section, Introduction 03 Introduction Module, Section Solving Equations In this section, you will learn to solve quadratic equations graphically, by factoring, and by applying
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction
Lesson : Creating and Solving Quadratic Equations in One Variale Prerequisite Skills This lesson requires the use of the following skills: understanding real numers and complex numers understanding rational
More informationGraphs and polynomials
1 1A The inomial theorem 1B Polnomials 1C Division of polnomials 1D Linear graphs 1E Quadratic graphs 1F Cuic graphs 1G Quartic graphs Graphs and polnomials AreAS of STud Graphs of polnomial functions
More informationPolynomial Functions of Higher Degree
SAMPLE CHAPTER. NOT FOR DISTRIBUTION. 4 Polynomial Functions of Higher Degree Polynomial functions of degree greater than 2 can be used to model data such as the annual temperature fluctuations in Daytona
More informationUnit 3. Expressions and Equations. 118 Jordan School District
Unit 3 Epressions and Equations 118 Unit 3 Cluster 1 (A.SSE.): Interpret the Structure of Epressions Cluster 1: Interpret the structure of epressions 3.1. Recognize functions that are quadratic in nature
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its
More informationHigher. Polynomials and Quadratics. Polynomials and Quadratics 1
Higher Mathematics Polnomials and Quadratics Contents Polnomials and Quadratics 1 1 Quadratics EF 1 The Discriminant EF Completing the Square EF Sketching Paraolas EF 7 5 Determining the Equation of a
More informationSECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION
2.25 SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION PART A: LONG DIVISION Ancient Example with Integers 2 4 9 8 1 In general: dividend, f divisor, d We can say: 9 4 = 2 + 1 4 By multiplying both sides
More informationa b a b ab b b b Math 154B Elementary Algebra Spring 2012
Math 154B Elementar Algera Spring 01 Stud Guide for Eam 4 Eam 4 is scheduled for Thursda, Ma rd. You ma use a " 5" note card (oth sides) and a scientific calculator. You are epected to know (or have written
More informationAlgebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.
C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each
More informationSection 3.3 Graphs of Polynomial Functions
3.3 Graphs of Polynomial Functions 179 Section 3.3 Graphs of Polynomial Functions In the previous section we eplored the short run behavior of quadratics, a special case of polynomials. In this section
More information1Number ONLINE PAGE PROOFS. systems: real and complex. 1.1 Kick off with CAS
1Numer systems: real and complex 1.1 Kick off with CAS 1. Review of set notation 1.3 Properties of surds 1. The set of complex numers 1.5 Multiplication and division of complex numers 1.6 Representing
More informationAlgebra Final Exam Review Packet
Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample  Find the degree of 5 7 yz Degree of a polynomial:
More informationLearning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. xcoordinate of vertex Axis of symmetry. yintercept.
Name: Hour: Algebra A Lesson:.1 Graphing Quadratic Functions Learning Targets: Term Picture/Formula In your own words: Quadratic Function Standard Form: Parabola Verte Ma/Min coordinate of verte Ais of
More informationMATH 111 Departmental Midterm Exam Review Exam date: Tuesday, March 1 st. Exam will cover sections and will be NONCALCULATOR EXAM.
MATH Departmental Midterm Eam Review Eam date: Tuesday, March st Eam will cover sections 9 +  and will be NONCALCULATOR EXAM Terms to know: quadratic function, ais of symmetry, verte, minimum/maimum
More informationMTH 65 WS 3 ( ) Radical Expressions
MTH 65 WS 3 (9.19.4) Radical Expressions Name: The next thing we need to develop is some new ways of talking aout the expression 3 2 = 9 or, more generally, 2 = a. We understand that 9 is 3 squared and
More informationExam 2 Review F15 O Brien. Exam 2 Review:
Eam Review:.. Directions: Completely rework Eam and then work the following problems with your book notes and homework closed. You may have your graphing calculator and some blank paper. The idea is to
More informationPolynomial and Synthetic Division
Chapter Polynomial and Rational Functions y. f. f Common function: y Horizontal shift of three units to the left, vertical shrink Transformation: Vertical each yvalue is multiplied stretch each yvalue
More informationVisit us at: for a wealth of information about college mathematics placement testing!
North Carolina Early Mathematics Placement Testing Program, 94. Multiply: A. 9 B. C. 9 9 9 D. 9 E. 9 Solution and Answer to Question # will be provided net Monday, 984 North Carolina Early Mathematics
More informationAlgebra II Notes Polynomial Functions Unit Introduction to Polynomials. Math Background
Introduction to Polynomials Math Background Previously, you Identified the components in an algebraic epression Factored quadratic epressions using special patterns, grouping method and the ac method Worked
More informationThe Final Exam is comprehensive but identifying the topics covered by it should be simple.
Math 10 Final Eam Study Guide The Final Eam is comprehensive ut identifying the topics covered y it should e simple. Use the previous eams as your primary reviewing tool! This document is to help provide
More informationd. 2x 3 7x 2 5x 2 2x 2 3x 1 x 2x 3 3x 2 1x 2 4x 2 6x 2 3. a. x 5 x x 2 5x 5 5x 25 b. x 4 2x 2x 2 8x 3 3x 12 c. x 6 x x 2 6x 6 6x 36
Vertices: (.8, 5.), (.37, 3.563), (.6, 0.980), (5.373, 6.66), (.8, 7.88), (.95,.) Graph the equation for an value of P (the second graph shows the circle with P 5) and imagine increasing the value of P,
More information3 Polynomial and Rational Functions
3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental
More informationBell Quiz 23. Determine the end behavior of the graph using limit notation. Find a function with the given zeros , 2. 5 pts possible.
Bell Quiz 23 2 pts Determine the end behavior of the graph using limit notation. 5 2 1. g( ) = 8 + 13 7 3 pts Find a function with the given zeros. 4. 1, 2 5 pts possible Ch 2A Big Ideas 1 Questions
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8
More informationab is shifted horizontally by h units. ab is shifted vertically by k units.
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential Function: a, 0, Graph of an
More informationUnit 2 Polynomial Expressions and Functions Note Package. Name:
MAT40S Mr. Morris Unit 2 Polynomial Expressions and Functions Note Package Lesson Homework 1: Long and Synthetic p. 7 #3 9, 12 13 Division 2: Remainder and Factor p. 20 #3 12, 15 Theorem 3: Graphing Polynomials
More informationModule 9: Further Numbers and Equations. Numbers and Indices. The aim of this lesson is to enable you to: work with rational and irrational numbers
Module 9: Further Numers and Equations Lesson Aims The aim of this lesson is to enale you to: wor with rational and irrational numers wor with surds to rationalise the denominator when calculating interest,
More informationSection 6.2 Long Division of Polynomials
Section 6. Long Division of Polynomials INTRODUCTION In Section 6.1 we learned to simplify a rational epression by factoring. For eample, + 3 10 = ( + 5)( ) ( ) = ( + 5) 1 = + 5. However, if we try to
More informationNumber Plane Graphs and Coordinate Geometry
Numer Plane Graphs and Coordinate Geometr Now this is m kind of paraola! Chapter Contents :0 The paraola PS, PS, PS Investigation: The graphs of paraolas :0 Paraolas of the form = a + + c PS Fun Spot:
More informationACTIVITY 14 Continued
015 College Board. All rights reserved. Postal Service Write your answers on notebook paper. Show your work. Lesson 11 1. The volume of a rectangular bo is given by the epression V = (10 6w)w, where w
More informationAlgebraic Functions, Equations and Inequalities
Algebraic Functions, Equations and Inequalities Assessment statements.1 Odd and even functions (also see Chapter 7)..4 The rational function a c + b and its graph. + d.5 Polynomial functions. The factor
More informationSection 5.5 Complex Numbers
Name: Period: Section 5.5 Comple Numbers Objective(s): Perform operations with comple numbers. Essential Question: Tell whether the statement is true or false, and justify your answer. Every comple number
More information3.1 Power Functions & Polynomial Functions
3.1 Power Functions & Polynomial Functions A power function is a function that can be represented in the form f() = p, where the base is a variable and the eponent, p, is a number. The Effect of the Power
More informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.3 Real Zeros of Polynomial Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Use long
More informationACCUPLACER MATH 0311 OR MATH 0120
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises
More informationf ( x ) = x Determine the implied domain of the given function. Express your answer in interval notation.
Test Review Section.. Given the following function: f ( ) = + 5  Determine the implied domain of the given function. Epress your answer in interval notation.. Find the verte of the following quadratic
More informationLesson #33 Solving Incomplete Quadratics
Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique
More informationMath3. Lesson 31 Finding Zeroes of NOT nice 3rd Degree Polynomials
Math Lesson  Finding Zeroes of NOT nice rd Degree Polynomials f ( ) 4 5 8 Is this one of the nice rd degree polynomials? a) Sum or difference of two cubes: y 8 5 y 7 b) rd degree with no constant term.
More informationMath 2412 Activity 2(Due by EOC Feb. 27) Find the quadratic function that satisfies the given conditions. Show your work!
Math 4 Activity (Due by EOC Feb 7) Find the quadratic function that satisfies the given conditions Show your work! The graph has a verte at 5, and it passes through the point, 0 7 The graph passes through
More informationSyllabus Objective: 2.9 The student will sketch the graph of a polynomial, radical, or rational function.
Precalculus Notes: Unit Polynomial Functions Syllabus Objective:.9 The student will sketch the graph o a polynomial, radical, or rational unction. Polynomial Function: a unction that can be written in
More informationLesson 10.1 Solving Quadratic Equations
Lesson 10.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with each set of conditions. a. One intercept and all nonnegative yvalues b. The verte in the third quadrant and no
More informationSchool of Business. Blank Page
Equations 5 The aim of this unit is to equip the learners with the concept of equations. The principal foci of this unit are degree of an equation, inequalities, quadratic equations, simultaneous linear
More informationTEKS: 2A.10F. Terms. Functions Equations Inequalities Linear Domain Factor
POLYNOMIALS UNIT TEKS: A.10F Terms: Functions Equations Inequalities Linear Domain Factor Polynomials Monomial, Like Terms, binomials, leading coefficient, degree of polynomial, standard form, terms, Parent
More information5. Determine the discriminant for each and describe the nature of the roots.
4. Quadratic Equations Notes Day 1 1. Solve by factoring: a. 3 16 1 b. 3 c. 8 0 d. 9 18 0. Quadratic Formula: The roots of a quadratic equation of the form A + B + C = 0 with a 0 are given by the following
More information4.3 Division of Polynomials
4.3 Division of Polynomials Learning Objectives Divide a polynomials by a monomial. Divide a polynomial by a binomial. Rewrite and graph rational functions. Introduction A rational epression is formed
More informationCOUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra
COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed
More informationUnit 4: Polynomial and Rational Functions
50 Unit 4: Polynomial and Rational Functions Polynomial Functions A polynomial function y p() is a function of the form p( ) a a a... a a a n n n n n n 0 where an, an,..., a, a, a0 are real constants and
More informationUnit 11  Solving Quadratic Functions PART TWO
Unit 11  Solving Quadratic Functions PART TWO PREREQUISITE SKILLS: students should be able to add, subtract and multiply polynomials students should be able to factor polynomials students should be able
More informationHere is a general Factoring Strategy that you should use to factor polynomials. 1. Always factor out the GCF(Greatest Common Factor) first.
1 Algera and Trigonometry Notes on Topics that YOU should KNOW from your prerequisite courses! Here is a general Factoring Strategy that you should use to factor polynomials. 1. Always factor out the GCF(Greatest
More information41 Graphing Quadratic Functions
41 Graphing Quadratic Functions Quadratic Function in standard form: f() a b c The graph of a quadratic function is a. y intercept Ais of symmetry coordinate of verte coordinate of verte 1) f ( ) 4 a=
More informationFinding Slope. Find the slopes of the lines passing through the following points. rise run
Finding Slope Find the slopes of the lines passing through the following points. y y1 Formula for slope: m 1 m rise run Find the slopes of the lines passing through the following points. E #1: (7,0) and
More informationBasic ALGEBRA 2 SUMMER PACKET
Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write them in standard form. You will
More informationExample 1: What do you know about the graph of the function
Section 1.5 Analyzing of Functions In this section, we ll look briefly at four types of functions: polynomial functions, rational functions, eponential functions and logarithmic functions. Eample 1: What
More informationPolynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
More information1. Division by a Monomial
330 Chapter 5 Polynomials Section 5.3 Concepts 1. Division by a Monomial 2. Long Division 3. Synthetic Division Division of Polynomials 1. Division by a Monomial Division of polynomials is presented in
More informationDefinition: Quadratic equation: A quadratic equation is an equation that could be written in the form ax 2 + bx + c = 0 where a is not zero.
We will see many ways to solve these familiar equations. College algebra Class notes Solving Quadratic Equations: Factoring, Square Root Method, Completing the Square, and the Quadratic Formula (section
More informationPRECALCULUS GROUP FINAL FIRST SEMESTER Approximate the following 13 using: logb 2 0.6, logb 5 0.7, 2. log. 2. log b
PRECALCULUS GROUP FINAL FIRST SEMESTER 008 Approimate the following 13 using: log 0.6, log 5 0.7, and log 7 0. 9 1. log = log log 5 =... 5. log 10 3. log 7 4. Find all zeros algeraically ( any comple
More informationSection 7.1 Objective 1: Solve Quadratic Equations Using the Square Root Property Video Length 12:12
Section 7.1 Video Guide Solving Quadratic Equations by Completing the Square Objectives: 1. Solve Quadratic Equations Using the Square Root Property. Complete the Square in One Variable 3. Solve Quadratic
More informationConceptual Explanations: Logarithms
Conceptual Eplanations: Logarithms Suppose you are a iologist investigating a population that doules every year. So if you start with 1 specimen, the population can e epressed as an eponential function:
More informationPolynomial and Synthetic Division
Chapter Polynomial Functions. f y. Common function: y Transformation: Vertical stretch each yvalue is multiplied by, then a vertical shift nine units upward f Horizontal shift of three units to the left,
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE.1 The Rectangular Coordinate Systems and Graphs. Linear Equations in One Variable.3 Models and Applications. Comple Numbers.5 Quadratic Equations.6 Other
More informationA. Incorrect! Apply the rational root test to determine if any rational roots exist.
College Algebra  Problem Drill 13: Zeros of Polynomial Functions No. 1 of 10 1. Determine which statement is true given f() = 3 + 4. A. f() is irreducible. B. f() has no real roots. C. There is a root
More informationAlgebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.
Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and
More informationAppendices. Appendix A.1: Factoring Polynomials. Techniques for Factoring Trinomials Factorability Test for Trinomials:
APPENDICES Appendices Appendi A.1: Factoring Polynomials Techniques for Factoring Trinomials Techniques for Factoring Trinomials Factorability Test for Trinomials: Eample: Solution: 696 APPENDIX A.1 Factoring
More informationComplex fraction:  a fraction which has rational expressions in the numerator and/or denominator
Comple fraction:  a fraction which has rational epressions in the numerator and/or denominator o 2 2 4 y 2 + y 2 y 2 2 Steps for Simplifying Comple Fractions. simplify the numerator and/or the denominator
More information4.5 Rational functions.
4.5 Rational functions. We have studied graphs of polynomials and we understand the graphical significance of the zeros of the polynomial and their multiplicities. Now we are ready to etend these eplorations
More informationCh. 9.3 Vertex to General Form. of a Parabola
Ch. 9.3 Verte to General Form Learning Intentions: of a Parabola Change a quadratic equation from verte to general form. Learn to square a binomial & factor perfectsquare epressions using rectangle diagrams.
More informationPolynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.
Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10
More informationLESSON #17  FACTORING COMMON CORE ALGEBRA II FACTOR TWO IMPORTANT MEANINGS
1 LESSON #17  FACTORING COMMON CORE ALGEBRA II In the study of algebra there are certain skills that are called gateway skills because without them a student simply cannot enter into many more comple
More informationInstructor Notes for Chapters 3 & 4
Algebra for Calculus Fall 0 Section 3. Complex Numbers Goal for students: Instructor Notes for Chapters 3 & 4 perform computations involving complex numbers You might want to review the quadratic formula
More informationFactor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.
TEKS 5.4 2A.1.A, 2A.2.A; P..A, P..B Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find
More informationMore Polynomial Equations Section 6.4
MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division
More informationSection 4.3: Quadratic Formula
Objective: Solve quadratic equations using the quadratic formula. In this section we will develop a formula to solve any quadratic equation ab c 0 where a b and c are real numbers and a 0. Solve for this
More informationMath 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have
Math 10 Final Eam Review 1. 4 5 6 5 4 4 4 7 5 Worked out solutions. In this problem, we are subtracting one polynomial from another. When adding or subtracting polynomials, we combine like terms. Remember
More informationSection 8.5. z(t) = be ix(t). (8.5.1) Figure A pendulum. ż = ibẋe ix (8.5.2) (8.5.3) = ( bẋ 2 cos(x) bẍ sin(x)) + i( bẋ 2 sin(x) + bẍ cos(x)).
Difference Equations to Differential Equations Section 8.5 Applications: Pendulums MassSpring Systems In this section we will investigate two applications of our work in Section 8.4. First, we will consider
More informationThe Mean Version One way to write the One True Regression Line is: Equation 1  The One True Line
Chapter 27: Inferences for Regression And so, there is one more thing which might vary one more thing aout which we might want to make some inference: the slope of the least squares regression line. The
More informationReady To Go On? Skills Intervention 61 Polynomials
6A Read To Go On? Skills Intervention 6 Polnomials Find these vocabular words in Lesson 6 and the Multilingual Glossar. Vocabular monomial polnomial degree of a monomial degree of a polnomial leading
More informationChapter Five Notes N P U2C5
Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have
More informationf 0 ab a b: base f
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential
More informationThe first property listed above is an incredibly useful tool in divisibility problems. We ll prove that it holds below.
1 Divisiility Definition 1 We say an integer is divisile y a nonzero integer a denoted a  read as a divides if there is an integer n such that = an If no such n exists, we say is not divisile y a denoted
More information