Slide 1 / 276 Slide 2 / 276 lgebra II Polynomials: Operations and Functions 2014-10-22 www.njctl.org Slide 3 / 276 Slide 4 / 276 Table of ontents click on the topic to go to that section Properties of Exponents Review Operations with Polynomials Review Special inomial Products inomial Theorem Factoring Polynomials Review ividing Polynomials Polynomial Functions nalyzing Graphs and Tables of Polynomial Functions Zeros and Roots of a Polynomial Function Writing Polynomials from its Given Zeros Properties of Exponents Review Return to Table of ontents This section is intended to be a brief review of this topic. For more detailed lessons and practice see lgebra 1. Slide 5 / 276 Slide 6 / 276 Goals and Objectives Students will be able to simplify complex expressions containing exponents. Why do we need this? Exponents allow us to condense bigger expressions into smaller ones. ombining all properties of powers together, we can easily take a complicated expression and make it simpler.
Slide 7 / 276 Properties of Exponents Slide 8 / 276 Product of Powers Power of Powers Power of a product Negative exponent Power of 0 Quotient of Powers Slide 9 / 276 Slide 10 / 276 1 Simplify: 50m 6 q 8 15m 6 q 8 50m 8 q 15. 5m 2 q 3 10m 4 q 5 Solution not shown Slide 11 / 276 Slide 12 / 276 3 ivide: 4 Simplify: Solution not shown Solution not shown
Slide 13 / 276 Slide 14 / 276 Sometimes it is more appropriate to leave answers with positive exponents, and other times, it is better to leave answers without fractions. You need to be able to translate expressions into either form. 5 Simplify. The answer may be in either form. Write with positive exponents: Write without a fraction: Solution not shown Slide 15 / 276 6 Simplify and write with positive exponents: Slide 16 / 276 When fractions are to a negative power, a short-cut is to invert the fraction and make the exponent positive. Try... Solution not shown Slide 17 / 276 Slide 18 / 276 Two more examples. Leave your answers with positive exponents. 7 Simplify and write with positive exponents: Solution not shown
Slide 19 / 276 Slide 20 / 276 Operations with Polynomials Review Return to Table of ontents This section is intended to be a brief review of this topic. For more detailed lessons and practice see lgebra 1. Slide 21 / 276 Slide 22 / 276 Vocabulary Review Goals and Objectives Students will be able to combine polynomial functions using operations of addition, subtraction, multiplication, and division. monomial is an expression that is a number, a variable, or the product of a number and one or more variables with whole number exponents. polynomial is the sum of one or more monomials, each of which is a term of the polynomial. Put a circle around each term: Slide 23 / 276 Slide 24 / 276 Polynomials can be classified by the number of terms. The table below summarizes these classifications.
Slide 25 / 276 Slide 26 / 276 Identify the degree of each polynomial: Polynomials can also be classified by degree. The table below summarizes these classifications. Slide 27 / 276 Polynomial Function Slide 28 / 276 rag each relation to the correct box: Polynomial Functions Not Polynomial Functions polynomial function is a function in the form where n is a nonnegative integer and the coefficients are real numbers. The coefficient of the first term, a n, is the leading coefficient. polynomial function is in standard form when the terms are in order of degree from highest to lowest. f(x) = For extra practice, make up a few of your own! Slide 29 / 276 To add or subtract polynomials, simply distribute the + or - sign to each term in parentheses, and then combine the like terms from each polynomial. Examples: (2a 2 +3a - 9) + (a 2-6a +3) (2a 2 +3a - 9) - (a 2-6a +3) Watch your signs...forgetting to distribute the minus sign is one of the most common mistakes students make!! Slide 30 / 276 losure: set is closed under an operation if when any two elements are combined with that operation, the result is also an element of the set. Is the set of all polynomials closed under - addition? - subtraction? Explain or justify your answer.
Slide 31 / 276 Slide 32 / 276 9 Simplify Slide 33 / 276 Slide 34 / 276 12 What is the perimeter of the following figure? (answers are in units, assume all angles are right) 2x - 3 x 2 +5x - 2 8x 2-3x + 4-10x + 1 Slide 35 / 276 To multiply a polynomial by a monomial, you use the distributive property of multiplication over addition together with the laws of exponents. Example: Simplify. -2x(5x 2-6x + 8) Multiplying Polynomials Slide 36 / 276 13 What is the area of the rectangle shown? (-2x)(5x 2 ) + (-2x)(-6x) + (-2x)(8) -10x 3 + 12x 2 + -16x -10x 3 + 12x 2-16x
Slide 37 / 276 Slide 38 / 276 14 15 Find the area of a triangle (= 1 / 2bh) with a base of 5y and a height of 2y + 2. ll answers are in square units. Slide 39 / 276 Slide 40 / 276 ompare multiplication of polynomials with multiplication of integers. How are they alike and how are they different? iscuss how we could check this result. = Is the set of polynomials closed under multiplication? Slide 41 / 276 To multiply a polynomial by a polynomial, distribute each term of the first polynomial to each term of the second. Then, add like terms. efore combining like terms, how many terms will there be in each product below? 3 terms x 5 terms 5 terms x 8 terms 100 terms x 99 terms Slide 42 / 276 16 What is the total area of the rectangles shown?
Slide 43 / 276 Slide 44 / 276 17 18 Slide 45 / 276 Slide 46 / 276 Example Part : town council plans to build a public parking lot. The outline below represents the proposed shape of the parking lot. Write an expression for the area, in square yards, of this proposed parking lot. Explain the reasoning you used to find the expression. From High School SS Flip ook Slide 47 / 276 Slide 48 / 276 Example Part : The town council has plans to double the area of the parking lot in a few years. They create two plans to do this. The first plan increases the length of the base of the parking lot by p yards, as shown in the diagram below. Write an expression in terms of x to represent the value of p, in feet. Explain the reasoning you used to find the value of p. Example Part : The town council s second plan to double the area changes the shape of the parking lot to a rectangle, as shown in the diagram below. an the value of z be represented as a polynomial with integer coefficients? Justify your reasoning.
Slide 49 / 276 Slide 50 / 276 20 Find the value of the constant a such that 2 4 6-6 Special inomial Products Return to Table of ontents Square of a Sum (a + b) 2 = (a + b)(a + b) = a 2 + 2ab + b 2 Slide 51 / 276 The square of a + b is the square of a plus twice the product of a and b plus the square of b. Example: Slide 52 / 276 Square of a ifference (a - b) 2 = (a - b)(a - b) = a 2-2ab + b 2 The square of a - b is the square of a minus twice the product of a and b plus the square of b. Example: Slide 53 / 276 Product of a Sum and a ifference (a + b)(a - b) = a 2 + -ab + ab + -b 2 = Notice the sum of -ab and ab a 2 - b 2 equals 0. The product of a + b and a - b is the square of a minus the square of b. Example: Slide 54 / 276 2 + = 2 2 +2 + Practice the square of a sum by putting any monomials in for and.
Slide 55 / 276 2 - = 2 2-2 + Practice the square of a difference by putting any monomials in for and. How does this problem differ from the last? Study and memorize the patterns!! You will see them over and over again in many different ways. Slide 56 / 276 + - = 2 2 - This very important product is called the difference of squares. Practice the product of a sum and a difference by putting any monomials in for and. How does this problem differ from the last two? 21 Slide 57 / 276 22 Simplify: Slide 58 / 276 Slide 59 / 276 Slide 60 / 276 23 Simplify: 24 Multiply:
Slide 61 / 276 hallenge: See if you can work backwards to simplify the given problem without a calculator. Slide 63 / 276 Slide 62 / 276 Problem is from: -PR Trina's Triangles lick for link for commentary and solution. lice and her friend Trina were having a conversation. Trina said "Pick any 2 integers. Find the sum of their squares, the difference of their squares and twice the product of the integers. These 3 numbers are the sides of a right triangle." Trina had tried this with several examples and it worked every time, but she wasn't sure this "trick" would always work. a. Investigate Trina's conjecture for several pairs of integers. oes it work? b. If it works, then give a precise statement of the conjecture, using variables to represent the chosen integers, and prove it. If not true, modify it so that it is true, and prove the new statement. c. Use Trina's trick to find an example of a right triangle in which all of the sides have integer length. all 3 sides are longer than 100 units, and the 3 side lengths do not have any common factors. Slide 64 / 276 The inomial Theorem is a formula used to generate the expansion of a binomial raised to any power. inomial Theorem inomial Theorem Return to Table of ontents ecause the formula itself is very complex, we will see in the following slides some procedures we can use to simplify raising a binomial to any power. Slide 65 / 276 Slide 66 / 276 What happens when you multiply a binomial by itself n times? Evaluate: n = 0 n = 1 n = 2 n = 3
Slide 67 / 276 Slide 68 / 276 Let's try another one: Expand (x + y) 4 25 The exponent of x is 5 on the third term of the expansion of. True False What will be the exponents in each term of (x + y) 5? Slide 69 / 276 Slide 70 / 276 26 The exponents of y are decreasing in the expansion of 27 What is the exponent of a in the fourth term of? True False Slide 71 / 276 Slide 72 / 276 Pascal's Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Row 0 Row 4 To get the next row, we start and end with 1, then add the two numbers above the next terms. Fill in the next 2 rows... One way to find the coefficients when expanding a polynomial raised to the n th power is to use the n th row of Pascal's Triangle.
Slide 73 / 276 Slide 74 / 276 28 ll rows of Pascal's Triangle start and end with 1 True 29 What number is in the 5th spot of the 6th row of Pascal's Triangle? False Slide 75 / 276 30 What number is in the 2nd spot of the 4th row of Pascal's Triangle? Slide 76 / 276 Now that we know how to find the exponents and the coefficients when expanding binomials, lets put it together. Expand Teacher Notes Slide 77 / 276 Slide 78 / 276 nother Example Expand: (In this example, 2a is in place of x, and 3b is in place of y.) Now you try! Expand:
Slide 79 / 276 Slide 80 / 276 31 What is the coefficient on the third term of the expansion of Slide 81 / 276 Slide 82 / 276 33 The binomial theorem can be used to expand True False Factoring Polynomials Review Return to Table of ontents Slide 83 / 276 Factoring Polynomials Review The process of factoring involves breaking a product down into its factors. Here is a summary of factoring strategies: erry Method to factor Step 1: alculate ac. Slide 84 / 276 Step 2: Find a pair of numbers m and n, whose product is ac, and whose sum is b. Step 3: reate the product. Step 4: From each binomial in step 3, factor out and discard any common factor. The result is your factored form. Example:
Slide 85 / 276 Slide 86 / 276 Example: More factoring review... Step 1: ac = -15 and b = -2 Step 2: find m and n whose product is -15 and sum is -2; so m = -5 and n = 3 Step 3: (ax + m)(ax + n) = (3x - 5)(3x + 3) Step 4: (3x + 3) = 3(x + 1) so discard the 3 Therefore, 3x 2-2x - 5 = (3x - 5)(x + 1) (In this unit, sum or difference of cubes is not emphasized.) Slide 87 / 276 Slide 88 / 276 34 Factor out the GF: 15m 3 n - 25m 2-15mn 3 35 Factor: x 2 + 10x + 25 15m(mn - 10m - n 3 ) 5m(3m 2 n - 5m - 3n 3 ) 5mn(3m 2-5m - 3n 2 ) 5mn(3m 2-5m - 3n) E 15mn(mn - 10m - n 3 ) (x - 5)(x - 5) (x - 5)(x + 5) (x + 15)(x + 10) (x - 15)(x - 10) E Solution not shown Slide 89 / 276 36 Factor: mn + 3m - 4n 2-12n (n - 3)(m + 4n) (n - 3)(m - 4n) (n + 4)(m - n) Not factorable E Solution not shown 37 Factor: 121m 2 + 100n 2 (11m - 10n)(11m + 10m) (121m - n)(m + 100n) (11m - n)(11m + 100n) Not factorable E Solution not shown Slide 90 / 276
38 Factor: 121m 2-100n 2 (11m - 10n)(11m + 10n) (121m - n)(m + 100n) (11m - n)(11m + 100n) Not factorable E Solution not shown Slide 91 / 276 39 Factor: 10x 2-11x + 3 (2x - 1)(5x - 3) (2x + 1)(5x + 3) (10x - 1)(x + 3) (10x - 1)(x - 3) E Solution not shown Slide 92 / 276 Slide 93 / 276 40 Which expression is equivalent to 6x 3-5x 2 y - 24xy 2 + 20y 3? Slide 94 / 276 41 Which expressions are factors of 6x 3-5x 2 y - 24xy 2 + 20y 3? Select all that apply. x 2 (6x - 5y) + 4y 2 (6x + 5y) x 2 (6x - 5y) + 4y 2 (6x - 5y) x 2 (6x - 5y) - 4y 2 (6x + 5y) x 2 (6x - 5y) - 4y 2 (6x - 5y) x 2 + y 2 6x - 5y 6x + 5y x - 2y E x + 2y From PR sample test From PR sample test Slide 95 / 276 Slide 96 / 276 42 The expression x 2 (x - y) 3 - y 2 (x - y) 3 can be written in the form (x - y) a (x +y), where a is a constant. What is the value of a? Write the expression x - xy 2 as the product of the greatest common factor and a binomial: etermine the complete factorization of x - xy 2 : From PR sample test From PR sample test
Slide 97 / 276 Slide 98 / 276 ivision of Polynomials Here are 3 different ways to write the same quotient: ividing Polynomials Return to Table of ontents Slide 99 / 276 Slide 100 / 276 Examples lick to Reveal nswer Slide 101 / 276 Slide 102 / 276 43 Simplify 44 Simplify
Slide 103 / 276 Slide 104 / 276 45 The set of polynomials is closed under division. True False Slide 105 / 276 Slide 106 / 276 Slide 107 / 276 Slide 108 / 276
Slide 109 / 276 Slide 110 / 276 Slide 111 / 276 Slide 112 / 276 Slide 113 / 276 Slide 114 / 276
Slide 115 / 276 Slide 116 / 276 46 Simplify. Slide 117 / 276 Slide 118 / 276 47 Simplify. Slide 119 / 276 Slide 120 / 276
Slide 121 / 276 Slide 122 / 276 52 If f (1) = 0 for the function,, what is the value of a? Slide 123 / 276 Slide 124 / 276 53 If f (3) = 27 for the function,, what is the value of a? Polynomial Functions Return to Table of ontents Slide 125 / 276 Slide 126 / 276 Goals and Objectives Students will be able to sketch the graphs of polynomial functions, find the zeros, and become familiar with the shapes and characteristics of their graphs. Why We Need This Polynomial functions are used to model a wide variety of real world phenomena. Finding the roots or zeros of a polynomial is one of algebra's most important problems, setting the stage for future math and science study.
Slide 127 / 276 Graphs of Polynomial Functions Features: ontinuous curve (or straight line) Turns are rounded, not sharp Which are polynomials? Slide 128 / 276 The Shape of a Polynomial Function The degree of a polynomial function and the coefficient of the first term affect: the shape of the graph, the number of turning points (points where the graph changes direction), the end behavior, or direction of the graph as x approaches positive and negative infinity. If you have Geogebra on your computer, click below to go to an interactive webpage where you can explore graphs of polynomials. Slide 129 / 276 Slide 130 / 276 Optional Spreadsheet ctivity See the spreadsheet activity on the unit page for this unit entitled "Exploration of the values of the terms of a polynomial". Explore the impact of each term by changing values of the coefficients in row 1. Slide 131 / 276 Slide 132 / 276 Take a look at the graphs below. These are some of the simplest polynomial functions, y = x n. Notice that when n is even, the graphs are similar. What do you notice about these graphs? What would you predict the graph of y = x 10 to look like? Notice the shape of the graph y = x n when n is odd. What do you notice as n increases? What do you predict the graph of y = x 21 would look like? For discussion: despite appearances, how many points sit on the x-axis?
Slide 133 / 276 End behavior means what happens to the graph as x and as x -. What do you observe about end behavior? Slide 134 / 276 These are polynomials of even degree. Observations about end behavior? Polynomials of Even egree Polynomials of Odd egree Positive Lead oefficient Negative Lead oefficient Slide 135 / 276 These are polynomials of odd degree. Positive Lead oefficient Negative Lead oefficient Slide 136 / 276 End ehavior of a Polynomial Lead coefficient is positive Lead coefficient is negative Left End Right EndLeft End Right End Polynomial of even degree Polynomial of odd degree Observations about end behavior? Slide 137 / 276 End ehavior of a Polynomial egree: even egree: even Lead oefficient: positive Lead oefficient: negative Slide 138 / 276 End ehavior of a Polynomial egree: odd egree: odd Lead oefficient: positive Lead oefficient: negative s x, f(x) s x -, f(x) In other words, the function rises to the left and to the right. s x, f(x) - s x -, f(x) - In other words, the function falls to the left and to the right. s s In other words, the function falls to the left and rises to the right. s s In other words, the function rises to the left and falls to the right.
Slide 139 / 276 54 etermine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. Slide 140 / 276 55 etermine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. odd and positive odd and negative even and positive even and negative odd and positive odd and negative even and positive even and negative Slide 141 / 276 56 etermine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. Slide 142 / 276 57 etermine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. odd and positive odd and positive odd and negative odd and negative even and positive even and positive even and negative even and negative Slide 143 / 276 Slide 144 / 276 Odd and Even Functions Odd functions not only have the highest exponent that is odd, but all of the exponents are odd. n even function has only even exponents. Note: a constant has an even degree ( 7 = 7x 0 ) Examples: Odd function Even function Neither f(x)=3x 5-4x 3 + 2x h(x)=6x 4-2x 2 + 3 g(x)= 3x 2 + 4x - 4 y = 5x y = x 2 y = 6x - 2 g(x)=7x 7 + 2x 3 f(x)=3x 10-7x 2 r(x)= 3x 5 +4x 3-2
Slide 145 / 276 Slide 146 / 276 Slide 147 / 276 n even function is symmetric about the y-axis. Slide 148 / 276 60 hoose all that apply to describe the graph. E F Odd egree Odd Function Even egree Even Function Positive Lead oefficient Negative Lead oefficient efinition of an Even Function Slide 149 / 276 61 hoose all that apply to describe the graph. Slide 150 / 276 62 hoose all that apply to describe the graph. Odd egree Odd egree Odd Function Odd Function Even egree Even egree Even Function Even Function E Positive Lead oefficient E Positive Lead oefficient F Negative Lead oefficient F Negative Lead oefficient
Slide 151 / 276 Slide 152 / 276 63 hoose all that apply to describe the graph. 64 hoose all that apply to describe the graph. Odd egree Odd egree Odd Function Odd Function Even egree Even egree Even Function Even Function E Positive Lead oefficient E Positive Lead oefficient F Negative Lead oefficient F Negative Lead oefficient Slide 153 / 276 Zeros of a Polynomial "Zeros" are the points at which the polynomial intersects the x-axis. They are called "zeros" because at each point f (x) = 0. nother name for a zero is a root. polynomial function of degreen has at MOST n real zeros. n odd degree polynomial must have at least one real zero. (WHY?) Zeros Slide 154 / 276 Relative Maxima and Minima polynomial function of degree n has at MOST n - 1 turning points, also called relative maxima and relative minima. These are points where the graph changes from increasing to decreasing, or from decreasing to increasing. Relative Maxima Relative Minima Slide 155 / 276 Slide 156 / 276 65 How many zeros does the polynomial appear to have? 66 How many turning points does the polynomial appear to have?
Slide 157 / 276 67 How many zeros does the polynomial appear to have? Slide 158 / 276 68 How many turning points does the graph appear to have? How many of those are relative minima? Slide 159 / 276 69 How many zeros does the polynomial appear to have? Slide 160 / 276 70 How many turning points does the polynomial appear to have? How many of those are relative maxima? Slide 161 / 276 71 How many zeros does the polynomial appear to have? Slide 162 / 276 72 How many relative maxima does the graph appear to have? How many relative minima?
Slide 163 / 276 Slide 164 / 276 polynomial function can be sketched by creating a table, plotting the points, and then connecting the points with a smooth curve. nalyzing Graphs and Tables of Polynomial Functions Return to Table of ontents x y -3 58-2 19-1 0 0-5 1-2 2 3 3 4 4-5 Look at the first term to determine the end behavior of the graph. In this case, the coefficient is negative and the degree is odd, so the function rises to the left and falls to the right. Slide 165 / 276 How many zeros does this function appear to have? x y -3 58-2 19-1 0 0-5 1-2 2 3 3 4 4-5 nswer x y -3 58-2 19-1 0 0-5 1-2 2 3 3 4 4-5 Slide 166 / 276 There is a zero at x = -1, a second between x = 1 and x = 2, and a third between x = 3 and x = 4. How can we recognize zeros given only a table? nswer Slide 167 / 276 Intermediate Value Theorem Given a continuous function f(x), every value between f(a) and f(b) exists. Slide 168 / 276 The Intermediate Value Theorem justifies the statement that there is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4. Let a = 2 and b = 4, then f(a)= -2 and f(b)= 4. For every x-value between 2 and 4 there exists a y-value, so there must be an x-value for which y = 0. x y -3 58-2 19-1 0 0-5 1-2 2 3 3 4 4-5
Slide 169 / 276 73 How many zeros of the continuous polynomial given can be found using the table? x y -3-12 -2-4 -1 1 0 3 1 0 2-2 3 4 4-5 nswer Slide 170 / 276 74 If the table represents a continuous function, between which two values of x can you find the smallest x-value at which a zero occurs? x y -3-2 -1 0 E 1 F 2 G 3 H 4-3 -12-2 -4-1 1 0 3 1 0 2-2 3 4 4-5 nswer Slide 171 / 276 Slide 172 / 276 75 How many zeros of the continuous polynomial given can be found using the table? 76 ccording to the table, what is the least value of x at which a zero occurs on this continuous function? x y -3 2-2 0-1 5 0 2 1-3 2 4 3 4 4-5 nswer -3-2 -1 0 E 1 F 2 G 3 x y -3 2-2 0-1 5 0 2 1-3 2 4 3 4 4-5 nswer H 4 Slide 173 / 276 Slide 174 / 276 Relative Maxima and Relative Minima How do we recognize the relative maxima and minima from a table? There are 2 relative maximum points at x = -1 and at x = 1. The relative maximum value appears to be -1 (the y-coordinate). There is a relative minimum at (0, -2). In the table, as x goes from -3 to 1, f(x) is decreasing. s x goes from 1 to 3, f(x) is increasing. nd as x goes from 3 to 4, f(x) is decreasing. The relative maxima and minima occur when the direction changes from decreasing to increasing, or from increasing to decreasing. x f(x) -3 5-2 1-1 -1 0-4 1-5 2-2 The y-coordinate indicates this change in direction as its value rises or falls. 3 2 4 0
Slide 175 / 276 Slide 176 / 276 Slide 177 / 276 Slide 178 / 276 77 t approximately what x-values does a relative minimum occur? -3 E 1 78 t about what x-values does a relative maximum occur? -3 E 1-2 F 2-2 F 2-1 G 3-1 G 3 0 H 4 0 H 4 Slide 179 / 276 Slide 180 / 276 79 t about what x-values does a relative minimum occur? x y -3 E 1-3 5-2 F 2-2 1-1 G -1-1 3 0-4 0 H 4 1-5 nswer 80 t about what x-values does a relative maximum occur? x y -3 E 1-3 5-2 F 2-2 1-1 G -1-1 3 0-4 0 H 4 1-5 nswer 2-2 2-2 3 2 3 2 4 0 4 0
Slide 181 / 276 Slide 182 / 276 81 t about what x-values does a relative minimum occur? x y -3 E 1-2 F 2-1 G 3 0 H 4-3 2-2 0-1 5 0 2 1-3 nswer 82 t about what x-values does a relative maximum occur? x y -3 E 1-2 F 2-1 G 3 0 H 4-3 2-2 0-1 5 0 2 1-3 nswer 2 4 2 4 3 4 3 5 4-5 4-5 Slide 183 / 276 Slide 184 / 276 Real Zeros of Polynomial Functions For a function f(x) and a real number a, if f (a) = 0, the following statements are equivalent: Zeros and Roots of a Polynomial Function x = a is a zero of the function f(x). x = a is a solution of the equation f (x) = 0. (x - a) is a factor of the function f(x). (a, 0) is an x-intercept of the graph of f(x). Return to Table of ontents Slide 185 / 276 Slide 186 / 276 The Fundamental Theorem of lgebra omplex Numbers If f (x) is a polynomial of degree n, where n > 0, then f (x) = 0 has n zeros including multiples and imaginary zeros. n imaginary zero occurs when the solution to f (x) = 0 contains complex numbers. Imaginary zeros are not seen on the graph. omplex numbers will be studied in detail in the Radicals Unit. ut in order to fully understand polynomial functions, we need to know a little bit about complex numbers. Up until now, we have learned that there is no real number, x, such that x 2 = -1. However, there is such a number, known as the imaginary unit, i, which satisfies this equation and is defined as. The set of complex numbers is the set of numbers of the form a + bi, where a and b are real numbers. When a = 0, bi is called a pure imaginary number.
Slide 187 / 276 The square root of any negative number is a complex number. For example, find a solution for x 2 = -9: Slide 188 / 276 rag each number to the correct place in the diagram. omplex Numbers Real Imaginary 9+6i 2/3 3i -11 2-4i -0.765 Slide 189 / 276 The number of the zeros of a polynomial, both real and imaginary, is equal to the degree of the polynomial. Slide 190 / 276 This 5th degree polynomial has 5 zeros, but only 3 of them are real. Therefore, there must be two imaginary. This is the graph of a polynomial with degree 4. It has four unique zeros: -2.25, -.75,.75, 2.25 Since there are 4 real zeros, there are no imaginary zeros. (4 in total - 4 real = 0 imaginary) (How do we know that this is a 5th degree polynomial?) Note: imaginary roots always come in pairs: if a + bi is a root, then a - bi is also a root. (These are called conjugates - more on that in later units.) Slide 191 / 276 vertex on the x-axis indicates a multiple zero, meaning the zeroccurs two or more times. Slide 192 / 276 What do you think are the zeros and their multiplicity for this function? 2 zeros each This is a 4th-degree polynomial. It has two unique real zeros: -2 and 2. These two zeros are said to have a multiplicity of two, which means they each occur twice. There are 4 real zeros and therefore no imaginary zeros for this function.
Slide 193 / 276 Notice the function for this graph. x - 1 is a factor two times, and x = 1 is a zero twice. x + 2 is a factor two times, and x = -2 is a zero twice. Therefore, 1 and -2 are zeros with multiplicity of 2. x + 3 is a factor once, and x = 3 is a zero with multiplicity of 1. Slide 194 / 276 83 How many real zeros does the 4th-degree polynomial graphed have? 0 1 2 3 E 4 F 5 Slide 195 / 276 84 o any of the zeros have a multiplicity of 2? Yes No Slide 196 / 276 85 How many imaginary zeros does this 7th degree polynomial have? 0 1 2 3 E 4 F 5 Slide 197 / 276 86 How many real zeros does the 3rd degree polynomial have? 0 1 2 3 E 4 F 5 Slide 198 / 276 87 o any of the zeros have a multiplicity of 2? Yes No
Slide 199 / 276 88 How many imaginary zeros does the 5th degree polynomial have? 0 1 2 3 E 4 F 5 Slide 200 / 276 89 How many imaginary zeros does this 4 th -degree polynomial have? 0 1 2 3 E 4 F 5 Slide 201 / 276 90 How many real zeros does the 6th degree polynomial have? 0 1 2 3 E 4 F 6 Slide 202 / 276 91 o any of the zeros have a multiplicity of 2? Yes No Slide 203 / 276 92 How many imaginary zeros does the 6th degree polynomial have? 0 1 2 3 E 4 F 5 Slide 204 / 276 Finding the Zeros from an Equation in Factored Form: Recall the Zero Product Property. If the product of two or more quantities or factors equals 0, then at least one of the quantities must equal 0.
Slide 205 / 276 Slide 206 / 276 So, if, then the zeros of are 0 and -1. So, if, then the zeros of are and. Slide 207 / 276 Slide 208 / 276 Slide 209 / 276 Slide 210 / 276 Find the zeros, including multiplicities, of the following polynomial. or or or or on't forget the ±!!
Slide 211 / 276 Slide 212 / 276 Slide 213 / 276 Slide 214 / 276 Slide 215 / 276 Slide 216 / 276
Slide 217 / 276 Slide 218 / 276 Slide 219 / 276 Slide 220 / 276 Slide 221 / 276 Find the zeros, showing the multiplicities, of the following polynomial. Slide 222 / 276 Find the zeros, including multiplicities, of the following polynomial. To find the zeros, you must first write the polynomial in factored form. or or or or or or or This polynomial has two distinct real zeros: 0 and 1. This is a 3 rd degree polynomial, so there are 3 zeros (count 1 twice). 1 has a multiplicity of 2. 0 has a multiplicity of 1. There are no imaginary zeros. This polynomial has 4 zeros. There are two distinct real zeros:, both with a multiplicity of 1. There are two imaginary zeros:, both with a multiplicity of 1.
Slide 223 / 276 105 How many zeros does the polynomial function have? 0 1 2 3 E 4 Slide 224 / 276 106 How many REL zeros does the polynomial equation have? 0 1 2 3 E 4 Slide 225 / 276 107 What are the zeros and their multiplicities of the polynomial function? Slide 226 / 276 108 Find the solutions of the following polynomial equation, including multiplicities. x = -2, mulitplicity of 1 x = -2, multiplicity of 2 x = 3, multiplicity of 1 x = 3, multiplicity of 2 E x = 0, multiplicity of 1 x = 0, multiplicity of 1 x = 3, multiplicity of 1 x = 0, multiplicity of 2 x = 3, multiplicity of 2 F x = 0, multiplicity of 2 Slide 227 / 276 109 Find the zeros of the polynomial equation, including multiplicities: Slide 228 / 276 110 Find the zeros of the polynomial equation, including multiplicities: x = 2, multiplicity 1 x = 2, multiplicity 2 x = -i, multiplicity 1 x = i, multiplicity 1 E x = -i, multiplcity 2 F x = i, multiplicity 2 2, multiplicity of 1 2, multiplicity of 2-2, multiplicity of 1-2, multiplicity of 2 E, multiplicity of 1 F -, multiplicity of 1
Slide 229 / 276 Find the zeros, showing the multiplicities, of the following polynomial. Slide 230 / 276 We are going to need to do some long division, but by what do we divide? To find the zeros, you must first write the polynomial in factored form. However, this polynomial cannot be factored using normal methods. What do you do when you are STUK?? The Remainder Theorem told us that for a function, f (x), if we divide f (x) by x - a, then the remainder is f (a). If the remainder is 0, then x - a if a factor of f (x). In other words, if f (a) = 0, then x - a is a factor of f (x). So how do we figure out what a should be???? We could use guess and check, but how can we narrow down the choices? Let Slide 231 / 276 The Rational Zeros Theorem: Slide 232 / 276 RTIONL ZEROS THEOREM Make list of POTENTIL rational zeros and test them out. with integer coefficients. There is a limited number of possible roots or zeros. Integer zeros must be factors of the constant term, a 0. Rational zeros can be found by writing and simplifying fractions where the numerator is an integer factor of a 0 and the denominator is an integer fraction of a n. Potential List: Hint: To check for zeros, first try the smaller integers -- they are easier to work with. Slide 233 / 276 Using the Remainder Theorem, we find that 1 is a zero: therefore (x -1) is a factor of the polynomial. Use POLYNOMIL IVISION to factor out. Slide 234 / 276 Find the zeros using the Rational Zeros Theorem, showing the multiplicities, of the following polynomial. Potential List: ± or or or or Hint: since all of the signs in the polynomial are +, only negative numbers will work. Try -3: ±1 This polynomial has three distinct real zeros: -2, -1/3, and 1, each with a multiplicity of 1. There are no imaginary zeros. -3 is a distinct zero, therefore (x + 3) is a factor. Use POLYNOMIL IVISION to factor out.
Slide 235 / 276 Slide 236 / 276 111 Which of the following is a zero of? or or or or x = -1 x = 1 x = 7 x = -7 This polynomial has two distinct real zeros: -3, and -1. -3 has a multiplicity of 2 (there are 2 factors of x + 3). -1 has a multiplicity of 1. There are no imaginary zeros. Slide 237 / 276 112 Find the zeros of the polynomial equation, including multiplicities, using the Rational Zeros Theorem Slide 238 / 276 113 Find the zeros of the polynomial equation, including multiplicities, using the Rational Zeros Theorem x = 1, multiplicity 1 x = 1, mulitplicity 2 x = 1, multiplicity 3 x = -3, multiplicity 1 E x = -3, multiplicity 2 F x = -3, multiplicity 3 x = -2, multiplicity 1 x = -2, multiplicity 2 Pull for ns wer x = -2, multiplicity 3 x = -1, multiplicity 1 E x = -1, multiplicity 2 F x = -1, multiplicity 3 Slide 239 / 276 Slide 240 / 276 114 Find the zeros of the polynomial equation, including multiplicities, using the Rational Zeros Theorem Pull for nswer x = 1, multiplicity 1 E x =, multiplicity 1 x = -1, multiplicity 1 F x =, multiplicity 1 x = 3, multiplicity 1 x = -3, multiplicity 1 G x =, multiplicity 1 H x =, multiplicity 1
Slide 241 / 276 Slide 242 / 276 116 Find the zeros of the polynomial equation. x = 2 x = -2 x =3 x = -3 E x = 3i F x = -3i G x = H x = - Writing a Polynomial Function from its Given Zeros Return to Table of ontents Slide 243 / 276 Slide 244 / 276 Write (in factored form) the polynomial function of lowest degree using the given zeros, including any multiplicities. Goals and Objectives Students will be able to write a polynomial from its given zeros. x = -1, multiplicity of 1 x = -2, multiplicity of 2 x = 4, multiplicity of 1 Work backwards from the zeros to the original polynomial. or or or For each zero, write the corresponding factor. or or or Slide 245 / 276 117 Write the polynomial function of lowest degree using the zeros given. x = -.5, multiplicity of 1 x = 3, multiplicity of 1 x = 2.5, multiplicity of 1 Slide 246 / 276 118 Write the polynomial function of lowest degree using the zeros given. x = 1/3, multiplicity of 1 x = -2, multiplicity of 1 x = 2, multiplicity of 1
Slide 247 / 276 119 Write the polynomial function of lowest degree using the zeros given. x = 0, multiplicity of 3 x = -2, multiplicity of 2 x = 2, multiplicity of 1 x = 1, multiplicity of 1 x = -1, multiplicity of 2 Slide 248 / 276 Write the polynomial function of lowest degree using the zeros from the given graph, including any multiplicities. x = -2 x = -1 x = 1.5 x = 3 x = -2 x = 1.5 x = -1 x = 3 E or or or Slide 249 / 276 120 Write the polynomial function of lowest degree using the zeros from the given graph, including any multiplicities. Slide 250 / 276 121 Write the polynomial function of lowest degree using the zeros from the given graph, including any multiplicities. E F Slide 251 / 276 Slide 252 / 276 Match each graph to its equation. y = x 2 + 2 y = (x-1)(x-2)(x-3) 2 y = (x-1)(x-2)(x-3) y = (x + 2) 2
Slide 253 / 276 Sketch Sketch the graph of f(x) = (x-1)(x-2) 2. fter sketching, click on the graph to see how accurate your sketch is. Slide 254 / 276 nalyzing Graphs using a Graphing alculator Enter the function into the calculator (Hit y= then type). heck your graph, then set the window so that you can see the zeros and the relative minima and maxima. (Look at the table to see what the min and max values of x and y should be.) Use the alc functions ( 2nd TRE ) to find zeros: Select 2: Zero Your graph should appear. The question "Left ound?" should be at the bottom of the screen. Use the left arrow to move the blinking cursor to the left side of the zero and press ENTER. The question "Right ound?" should be at the bottom of the screen. Use the right arrow to move the blinking cursor to the right side of the zero and press ENTER. The question "Guess?" should be at the bottom of the screen. Press ENTER again, and the coordinates of the zero will be given. Slide 255 / 276 Slide 256 / 276 Finding Minima and Maxima Use the alc functions (2nd TRE) to find relative min or max: Select 3: minimum or 4: maximum. Your graph should appear. The question "Left ound?" should be at the bottom of the screen. Use the left arrow to move the blinking cursor to the left side of the turning point and press ENTER. The question "Right ound?" should be at the bottom of the screen. Use the right arrow to move the blinking cursor to the right side of the turning point and press ENTER. The question "Guess?" should be at the bottom of the screen. Press ENTER again, and the coordinates of the min or max will be given. Use a graphing calculator to find the zeros and turning points of Note: The calculator will give an estimate. Rounding may be needed. Slide 257 / 276 Use a graphing calculator to find the zeros and turning points of Slide 258 / 276 Sketch the graph of f(x) = (x-1)(x+1)(x-2)(x+2)(x-3)(x+3)(x-4). fter sketching, click on the graph to see how accurate your sketch is.
Slide 259 / 276 The product of 4 positive consecutive integers is 175,560. Slide 260 / 276 n open box is to be made from a square piece of cardboard that measures 50 inches on a side by cutting congruent squares of sidelength x from each corner and folding the sides. Write a polynomial equation to represent this problem. 1. Write the equation of a polynomial function to represent the volume of the completed box. Use a graphing utility or graphing calculator to find the numbers. Hint: set your equation equal to zero, and then enter this equation into the calculator. How could you use a calculator and guess and check to find the answer to this problem? 50 x x 2. Use a graphing calculator or graphing utility to create a table of values for the height of the box. (onsider what the domain of x would be.) Use the table to determine what height will yield the maximum volume. 3. Look at the graph and calculate the maximum volume within the defined domain. oes this answer match your answer above? (Use the table values to determine how to set the viewing window.) Slide 261 / 276 n engineer came up with the following equation to represent the height, h(x), of a roller coaster during the first 300 yards of the ride: h(x) = -3x 4 + 21x 3-48x 2 + 36x, where x represents the horizontal distance of the roller coaster from its starting place, measured in 100's of yards. Using a graphing calculator or a graphing utility, graph the function on the interval 0 < x < 3. Sketch the graph below. Slide 262 / 276 For what values of x is the roller coaster 0 yards off the ground? What do these values represent in terms of distance from the beginning of the ride? oes this roller coaster look like it would be fun? Why or why not? Verify your answers above by factoring the polynomial h(x) = -3x 4 + 21x 3-48x 2 + 36x (erived from ( Slide 263 / 276 Slide 264 / 276 How do you think the engineer came up with this model? onsider the function f(x) = x 3-13x 2 + 44x - 32. Use the fact that x - 4 is a factor to factor the polynomial. Why did we restrict the domain of the polynomial to the interval from 0 to 3? What are the x-intercepts for the graph of f? In the real world, what is wrong with this model at a distance of 0 yards and at 300 yards? t which x-values does the function change from increasing to decreasing and from decreasing to increasing?
Slide 265 / 276 How can we tell if a function is positive or negative on an interval between x-intercepts? Given our polynomial f(x) = x 3-13x 2 + 44x - 32... Slide 266 / 276 123 onsider the function f (x)=(2x -1)(x + 4)(x - 2). What is the y-intercept of the graph of the function in the coordinate plane? When x < 1, is the graph above or below the x-axis? When 1 < x < 4, is the graph above or below the x-axis? When 4 < x < 8, is the graph above or below the x-axis? When x > 8, is the graph above or below the x-axis? From PR sample test Slide 267 / 276 onsider the function f (x)=(2x -1)(x + 4)(x - 2). For what values of x is f (x) >0? Use the line segments and endpoint indicators to build the number line that answers the question. Slide 268 / 276 124 onsider the function f (x)=(2x -1)(x + 4)(x - 2). What is the end behavior of the graph of the function? From PR sample test From PR sample test Slide 269 / 276 125 onsider the function f (x)=(2x -1)(x + 4)(x - 2). How many relative maximums does the function have? Slide 270 / 276 How many relative maxima and minima? none one two three f(x) = (x+1)(x-3) g(x) = (x-1)(x+3)(x-4) h(x) =x (x-2)(x-5)(x+4) f(x) g(x) h(x) egree: # x-intercepts: # turning points: nswer From PR sample test Observations:
Slide 271 / 276 How many relative maxima and minima? Slide 272 / 276 Increasing and ecreasing Given a function f whose domain and range are subsets of the real numbers and I is an interval contained within the domain, the function is called increasing on the interval if f (x 1) < f (x 2) whenever x 1 < x 2 in I. egree: f(x) g(x) h(x) nswer It is called decreasing on the interval if f (x 1) > f (x 2) whenever x 1 < x 2 in I. Restate this in your own words: # x-intercepts: # turning points: Observations: Slide 273 / 276 Slide 274 / 276 Mark on this graph and state using inequality notation the intervals that are increasing and those that are decreasing. 126 Select all of the statements that are true based on the graph provided: The degree of the function is even. There are 4 turning points. The function is increasing on the interval from x = -1 to x = 2.4. The function is increasing when x < -1. E x - 2 and x + 3 are factors of the polynomial that defines this function. Slide 275 / 276 127 Given the function f (x) = x 7-4x 5 - x 3 + 4x. Which of the following statements are true? Select all that apply. s x, f (x). Slide 276 / 276 For each function described by the equations and graphs shown, indicate whether the function is even, odd, or neither even nor odd: f(x)=3x 2 g(x)=-x 3 + 5 There are a maximum of 6 real zeros for this function. x = -1 is a solution to the equation f (x) = 0. The maximum number of relative minima and maxima for this function is 7. h(x) f(x) g(x) h(x) k(x) Even Odd Neither k(x) nswer k(x) From PR sample test