NC Math 3 Modelling with Polynomials
|
|
- Nickolas Jordan
- 6 years ago
- Views:
Transcription
1 NC Math 3 Modelling with Polynomials
2
3 Introduction to Polynomials; Polynomial Graphs and Key Features Polynomial Vocabulary Review Expression: Equation: Terms: o Monomial, Binomial, Trinomial, Polynomial Degree: o Constant, Linear, Quadratic, Cubic, Quartic POLYNOMIAL NUMBER OF TERMS CLASSIFICATION BY TERMS DEGREE CLASSFICATION BY DEGREE f(x) = 5 g(x) = 4x 3 p(x) = 2x 5 w(x) = x 4 4x + 2 y = 4x 2 + x + 9 h(x) = 4x 3 + x 2 9x
4 The number k is said to be a zero of a polynomial if f(k) = 0. k is often referred to as the root or solution If k is a real number, then f(k) = 0 means that the graph crosses the x-axis at that value. k can also be referred to as an x-intercept Check out the graphs below and identify any values that represent a zero/solution/root. A. B. factored equation: factored equation: C. D. factored equation: factored equation: 3
5 WATCH OUT! Multiplicities of Zeros If c is a zero of the function P and the corresponding factor (x c) occurs exactly m times in the factorization of P then we say that c is a zero of multiplicity m. One can show that the graph of P crosses the x-axis at c if the multiplicity m is odd and does not cross the x-axis if m is even. 4
6 Polynomial Degrees, Roots, Factored Form and Turning Points Degree Number of Unique Real Roots (indicate if a root has a multiplicity greater than 1) Number of Non-Real Roots Sketch a Graph Possible Equation of the graph in Factored Form (Only if all roots are real) Number of Turning Points (maxima and minima) (multiplicity of 2) 0 5
7 Degree Number of Unique Real Roots (indicate if a root has a multiplicity greater than 1) Number of Non-Real Roots Sketch a Graph Possible Equation of the graph in Factored Form (Only if all roots are real) Number of Turning Points (maxima and minima) (with a multiplicity of 3) 3 1 (multiplicity of 2) 1 6
8 Degree Number of Unique Real Roots (indicate if a root has a multiplicity greater than 1) Number of Non-Real Roots Sketch a Graph Possible Equation of the graph in Factored Form (Only if all roots are real) Number of Turning Points (maxima and minima) (multiplicity of 2) 7
9 Degree Number of Unique Real Roots (indicate if a root has a multiplicity greater than 1) Number of Non-Real Roots Sketch a Graph Possible Equation of the graph in Factored Form (Only if all roots are real) Number of Turning Points (maxima and minima) 2 (each having a multiplicity of 2) 0 4 (1 having a multiplicity of 2, and the other two are unique) 0 Summary Questions: 1. What is the relationship between degree and number of roots? 2. What is the relationship between degree and number of turning points? 3. What is the relationship between the factored form of the equation and the x-intercepts? 4. Why do you think that non-real roots always come in pairs? (**Hint** Think about the quadratic formula, why can you not get only one non-real answer when you use the quadratic formula?) 8
10 The high and low points on a graph are called the extrema of the function. An extremum that is higher or lower than any other points nearby is called a relative extremum. A relative extremum (the plural of extremum is extrema) that is higher than points nearby is called relative maximum. A relative extremum that is lower than points nearby is called a relative minimum. A function s absolute extremum occurs at the highest or lowest point on a function. The highest point on a function is called the absolute maximum and the lowest point on a function is called the absolute minimum. Can you identify these in the graphs on the previous page? The end behavior of a polynomial is a description of what happens as x becomes large in the positive or negative direction. For any polynomial, the end behavior is determined by the term that contains the highest power of x, because when x is large, the other terms are relatively insignificant in size. Given the graph of the polynomial below - State the intervals where the graph is increasing/decreasing (think slope ) - State the intervals where the graph is positive/negative (above/below the x-axis) 9
11 Polynomial Graphs Homework 1. Fill in the missing information. Polynomial Function Name (degree) Name (terms) End Behavior f(x) = 3x 2 5 y = x 4 + 6x 1 g(x) = 6x h(x) = 5x 2 2x 3 + 7x 3 2. Identify the zeros of each function below. Be sure to state any multiplicity. 10
12 3. Use the given information to complete the missing columns. Table of Values Graph Key features of the function X Y X Y The y-intercept is (0, 7). The zeros are located at x = 4 and x = 7. There is a relative minimum at (- 5.5, 1.5) and at (5.5, -2.5). A relative maximum is located at (-1, 8.5). The polynomial is quartic. 4. Given the graph, state the intervals where the graph is increasing/decreasing and where the graph is positive/negative. 11
13 Multiple Representations of Functions: Table & Graph 1. Write the equation for your assigned polynomial function below. 2. Use the given polynomial function to complete the table. Include values from various parts of the function. (You may want to use your calculator to graph the function first to get an idea of what the graph looks like.) x y 3. Graph your function on the grid provided. Use a marker to help the function stand out. Include the equation on the graph page. (Make sure that the important features of the function are graphed.) 4. Pair Share: Compare your table and graph with your partner. Discuss some of the similarities and differences that you found between your two functions. Note any interesting similarities or differences between polynomial functions and functions that been discussed previously. Write down one thought. 12
14 Equation: 13
15 Multiple Representations of Functions: Reflection Part I: Number of Zeros: 1. Record at least two similarities and differences between the functions that have the same number of zeros. Include examples. Similarities Differences 2. What conclusions can be made about polynomial functions and the number of zeros they have? Are there any special cases? If so, what conclusions can be made about the special cases? Part II: End Behavior: 3. Record at least two similarities and differences between the functions that have the same end behavior. Include examples. Similarities Differences 4. What conclusions can be made about polynomial functions and their end behavior? Are there any special cases? If so, what conclusions can be made about the special cases? 14
16 Polynomial Graphs and Zeros The degree of a polynomial function gives a lot of information y = ax y = ax 2 y = ax 3 y = ax 4 y = ax 5 Type of Polynomial Function LINEAR QUADRATIC CUBIC QUARTIC QUINTIC Domain Range Maximum number of solutions/zeros (this is equal to the degree of the polynomial) Maximum number of turns in the graph (this is one less than the degree of the polynomial) Possible shape of the graph Positive a Negative a End behavior Positive a Negative a 15
17 Polynomial Graph Matching Homework Part 1: Match each equation to its graph, zeros and end behavior 1) y =.8x 2 3) y = -2x + 3x - 8 2) y = 4.5x x 4 + 3x 3-5x 2 + 6x ) y = (x+2)(x-3)(x+5)(x-8) 5) y = x 3-8x x ) y = -2x 3 + 4x 2 + 4x + 15 A. B. C. D. E. F. # of zeros: End Behavior: 4 real roots 3 real roots 1 real root, 2 imaginary 1 real root, 4 imaginary roots 1 real root roots (, - ) (, ) (-, ) 2 real roots (, - ) (-, ) (, ) Write the equation Graph letter # of zeros End beavior Created by: Cortella/Taylor,
18 Zeros of a Polynomial Function Part 1: Look at the graph and state the x-intercepts; watch out for repeated roots! x-intercepts: equation: x-intercepts: equation: x-intercepts: equation: Part 2: Use the calculator to find any exact roots. Zeros: Factored form: Zeros: Factored form: Zeros: Factored form: 17
19 Graph each function. 1. y = -2( x 2-9) ( x + 4) 2. y = (x 2-4)(x+3) 3. y = -1(x 2-9)(x 2-4) 4. y = 1 4 ( x + 2) ( x -1)2 5. y = 1 5 ( x - 3 ( )2 x +1) 2 6. y = ( x +1) 3 ( x - 4) 18
20 Math 3 Polynomial Parent Functions Linear Quadratic Cubic Quartic Function Equation State the type of Function Sketch the function Words: The graph moved (compare to the parent function) y = x 3 3 y = (x + 5) 2 y = ( 3x + 1) 4 y = x
21 Function Equation State the type of Function Sketch the function Words: The graph moved (compare to the parent function) y = (x + 4) 3 y = x 4 2 y = ( 1 4 x + 1) y = 2x + 5 y = 3x 2 20
22 Function Equation Parent Name Graph the function Words: The graph moved (compare to the parent function) y = ( 2x 1) y = (x + 1) General Form of a function f(x) = af(bx h) + k Summarize the different types of transformations When a > 1: When 0 < a < 1: When a is negative: When b > 1: When 0 < b < 1: When b is negative: When h is added: When h is subtracted: When k is added: When k is subtracted: 21
23 Polynomial Transformations: Check for Understanding 1. Write an equation that will move the graph of the function y=x 4 right 4 units and reflect over the x-axis. 2. The equation y = (x+3) 2 2 moves the parent function y = x 2 right 3 units and down 2 units. True or False 3. Write an equation that will move the graph of the function y = x 3 down 7 units with a horizontal stretch of The equation y = (x-8) moves the parent function y = x 2 right 8 units and down 5 units. True or False 5. Write an equation that will move the graph of the function y=x 4 left 2 units and up 6 units with a reflection across the y-axis. 6. Which equation will shift the graph of y = x 2 left 5 units and up 6 units? a. y = (x+6) 2-5 b. y = (x+5) 2-6 c. y = (x+5) 2 +6 d. y = (x-5) Write an equation that will move the graph of the function y=x 4 right 3 units up 2 units with a vertical stretch by 1/2. 8. Which equation will shift the graph of y = x 2 right 8 units and down 4 units? a. y = (x+8) 2-4 b. y = (x+4) 2-8 c. y = (x-4) 2 +8 d. y = (x-8)
24 Transformations of Polynomial Functions Homework 1. Write the equation for the graph of function g(x), obtained by shifting the graph of f (x) = x² three units left, stretching the graph vertically by a factor of two, reflecting that result over the x-axis, and then translating the graph up four units. 2. Describe the transformations that would produce the graph of the second function from the graph of the first function. a. f(x) = x 2 becomes f(x) = (x 3) b. f(x) = x 3 becomes f(x) = 3x 3 1 c. f(x) = x 4 becomes f(x) = 1 2 (x + 1)4 3 d. f(x) = x 2 becomes f(x) = 2(3x 2) Write the equation for the graph of function g(x), obtained by shifting the graph of f (x) = x 4 two units right and up four units
25 Polynomial Equations and Models 24
26 25
27 Polynomial Word Problems Practice and Homework 1. At the ruins of Caesarea, archaeologists discovered a huge hydraulic concrete block with a volume of 945 cubic meters. The block s dimensions are x meters high by 12x 15 meters long by 12x 21 meters wide. What is the height of the block? 2. You are designing a chocolate mold shaped like a hollow rectangular prism for a candy manufacturer. The mold must have a thickness of 1 cm in all dimensions. The mold s outer dimensions should also be in the ratio 1:3:6. What should the outer dimensions of the mold be if it is to hold 112 cubic centimeters of chocolate? 3. A manufacturer wants to build a rectangular stainless steel tank with a holding capacity of 670 gallons, or about cubic feet. The tank s walls will be one half inch thick and about 6.42 cubic feet of steel will be used for the tank. The manufacturer wants the outer dimensions of the tank to be related as follows: The width should be 2 feet less than the length The height should be 8 fee more than the length What should the outer dimensions of the tank be? (HINT: Volume of steel = Volume outside volume inside) 4. From 1985 to 2003, the total attendance A (in thousands) at NCAA women s basketball games and the number T of NCAA women s basketball teams can be modeled by A = 1.95x x 2 188x and T = 14.8x where x is the number of years since Compare and contrast the two functions. Find the attendance and number of teams for the year Suppose you have 250 cubic inches of clay with which to make a sculpture shaped as a rectangular prism. You want the height and width each to be 5 inches less than the length. What should the dimensions of the prism be if you want to use all of your clay? 6. The price p (in dollars) that a radio manufacturer is able to charge for a radio is given by p = 40 4x 2 where x is the number of radios produced in millions. It costs the company $15 to make a radio. a) Write an expression for the company s total revenue in terms of x b) Write a function for the company s profit P by subtracting the total cost to make x radios from the expression in part a c) Currently the company produces 1.5 million radios and makes a profit of $24,000,000. What lesser number of radios can the company produce to make the same profit? 26
28 7. CHALLENGE: The profit P (in millions of dollars) for a DVD manufacturer can be modeled by P = 6x x where x is the number of DVDs produced (in millions). Show that 2 million DVDs is the only production level for the company that yields a profit of $96,000, A platform shaped like a rectangular prism has dimensions (x 2) feet by (3 2x) feet by (3x + 4) feet. Explain why the volume of the platform cannot be 7/3 cubic feet. BONUS 27
29 WRITING EQUATIONS OF POLYNOMIALS Write the equation from the graph:
30 Polynomial Long Division Ex 1: (3x 3 5x x 3) 3x + 1 Ex 2: (2x 3 9x ) (2x 5) and the answer is: and the answer is:copyright Elizabeth All Rights Reserved The steps in the process of long division were: 1) Divide (first term into first term) 2) Multiply (use all terms) 3) Subtract 4) Bring Down 5) REPEAT Dividing Polynomials - EXAMPLES Dividing by a monomial 1. (-30x 3 y + 12x 2 y 2 18x 2 y) (-6x 2 y) 29
31 Divide using Long Division 2. (6x 2 x 7) (3x + 1) 3. (4x 2 2x + 6)(2x 3) (4x 3 8x 2 + 3x 8) (2x 1) 5. (2x 3 3x 2 18x 8) (x 4) 6. (2x 4 + 3x 3 + 5x -1) (x 2-2x + 2) 30
32 Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x a). It can be used in place of the standard long division algorithm. Example: (2x 4 3x 3-5x 2 + 3x + 8) (x-2) All the variables and their exponents are removed, leaving only a list of the coefficients: 2, -3, -5, 3, 8. (Note: if a power of x is missing from the polynomial, a term with that power and a zero coefficient must be inserted into the correct position in the polynomial.) These numbers form the dividend. We form the divisor for the synthetic division using only the constant term (2) of the linear factor (x-2). (Note: If the divisor were (x+2), we would use a divisor of -2.) The numbers representing the divisor and the dividend are placed into a division-like configuration. First, bring down the "2" that indicates the leading coefficient: Multiply by the number on the left, and carry the result into the next column: Add down the column: Multiply by the number on the left, and carry the result into the next column: Add down the column: Multiply by the number on the left, and carry the result into the next column: Add down the column: 31
33 Multiply by the number on the left, and carry the result into the next column: Add down the column for the remainder: The completed division is: All numbers except the last become the coefficients of the quotient polynomial. Since we started with a 4 th degree polynomial and divided it by a linear term, the quotient is a 3 rd degree polynomial. The last entry in the result list (2) is the remainder. The quotient and remainder can be combined into one expression: 2x 3 + x 2 3x /(x-2) (Note that no division operations were performed to compute the answer to this division problem.) The steps in the process of synthetic division were: 1) Bring Down 2) Multiply 3) Add 4) REPEAT from 2) Divide using Synthetic Division 7. (2x 2 + 3x 4) (x 2) 8. (x 4 3x 3 + 5x 6) (x + 2) 9. (2x 3 + 4x 6) (x + 3) 10. (x 4 2x 3 + 6x 2 8x + 10) (x + 2) (HONORS) 11. (6x 4 x 3 + 3x + 5) / (2x + 1) 32
34 Synthetic Division when the coefficient of x in the divisor 1 HONORS Divide: Step 1: Factor out the coefficient of x in the denominator. ( 4x3 8x 2 x + 5 2(x 1 2 ) ) 1 8x 2 x (4x3 x 1 ) 2 Step 2:, Set up the synthetic division, ignoring the ½ that was factored out. Step 3: Once the problem is set up correctly, bring the leading coefficient (first number) straight down. Step 4: Multiply the number in the division box with the number you brought down and put the result in the next column. Step 5: Add the two numbers together and write the result in the bottom of the row. Step 6: Repeat steps 3 and 4 until you reach the end of the problem. Step 7: Multiply everything by the ½ which was factored out in Step (4x2 6x x 1 ) 2 2x 2 3x x 1 is the final answer 33
35 Long Division Practice Divide each of the polynomials using long division. 1. (4x 2 9) (2x + 3) 2. (x 2 4) (x + 4) 3. (2x 2 + 5x 3) (x + 3) 4. (2x 2 + 5x 3) (x 3) 5. x x 4 9 x 2 1 x Synthetic Division Practice 7. (3x 2 13x 10) (x 5) 8. (3x 2 13x 10) (x + 5) 9. (11x + 20x x 3 + 2) (3x + 2) 10. (12x x + 20x 2 ) (2x + 1) 34
36 Polynomial Division Worksheet Divide using Long Division 1. (3y 3 + 2y 2 32y + 2) / (y 3) 3. (2c 3 3c 2 + 3c 4) / (c 2) 2. (2b 3 + b 2 2b + 3) / (b + 1) 4. (3x 3 2x 2 + 2x 1) / (x 1) Divide using Synthetic Division 5. (t 4 2t 3 + t 2 3t + 2) / (t 2) 10. (x 3 + 2x 2 5x 6) / (x 2) 6. (3r 4 6r 3 2r 2 + r 6) / (r + 1) 11. (x 3 + 3x 2 7x + 1) / (x 1) 7. (z 4 3z 3 z 2 11z 4) / (z 4) 12. (n 4 8n n + 105) / (n 5) 8. (2b 3 11b b + 9) / (b 3) 13. (2x 4 5x 3 + 2x 3) / (x 1) 9. (6s 3 19s 2 + s + 6) / (s 3) 14. (z 5 6z 3 + 4x 2 3) / (z 2) Divide using long division: 15. (4s 4 5s 2 + 2s + 3) / (2s 1) 16. (2x 3 3x 2 8x + 4) / (2x + 1) Divide using Synthetic Division to factor completely 17. (2x 3 + x 2 22x + 24) / (2x 3) 18. (6j 3 19j 2 + j + 6) / (3j 2) 35
37 Remainder Theorem Divide p(x) = x 3 7x 6 by the linear term (x-4) Therefore: x 3 7x 6 x 4 = Multiply both sides by (x-4): Since p(x) = x 3 7x 6 What is the value of p(4)? What conclusion can you make about the remainder? 36
38 Remainder Theorem The value of the polynomial p(x) at x=a is the same as the remainder you get when you divide the polynomial by x-a. To evaluate a polynomial p(x) at x=a, use synthetic division to divide the polynomial by x=a. The remainder is p(a) Use the Remainder Theorem and Synthetic Division do find f(4) where f(x) = x 3 + 8x The remainder theorem tells us that if we divide f(x) by (x-4) the remainder will be equal to f(4). Factor Theorem p(a) = 0 if and only if x-a is a factor of p(x). If you divide a polynomial by x=a and get a zero remainder, then not only is x=a a zero of the polynomial, but x-a is a factor of the polynomial. Determine whether x+4 is a factor of each polynomial A. f(x) = x 2 + 6x + 8 B. f(x) = x 3 + 3x 2 6x 7 37
39 Use the Remainder Theorem 1) Is (x 1) a factor of x 3 + 2x 2 2x 1? 2) Is (x + 2) a factor of 4x x + 10? 3) What is the remainder when 3x x 2 + x 6 is divided by x + 3 4) Is (x 2) a factor of 4x x + 10? 5) What is the remainder when 3x x 2 + x 6 is divided by x 1? Find the zeros using the given information 1) Find all the zeros of f(x) = x 3 4x 2 + x + 6 given that x + 1 is a factor. 2) Solve for all the solutions of 2x 3 5x 2 + x + 2 = 0 given that 2 is a solution. 3) Find all the zeros of g(x) = 2x 3 + 3x 2 + 8x + 12 if 3 is a root. 2 38
40 Zeros & Remainder Theorem, Factor Theorem Worksheet #1-6 Find the zeros of the polynomials and state the multiplicity of each zero: 1. f(x) = ( x + 4 ) 3 ( 3x 4 ) 2. f(x) = 2x 5 8x 4 10x 3 3. f(x) = ( 9x 2 25) 4 ( x ) 4. f(x) = (x 2 + x 2) 2 (x 2 4) 5. f(x) = x (x + 2) 3 (x 5) 6. f(x) = 2x (x+3) 2 (4x 1) #7&8 Write a polynomial equation in standard from having the given roots: 7. 2, 3, multiplicity 2, multiplicity 2, 1, 2 #10-15 Use synthetic division to show that c is a zero of f(x). 10. f(x) = 3x 4 + 8x 3 2x 2 10x + 4 ; c = f(x) = 4x 3 9x 2 8x 3 ; c = f(x) = 2x 3 + 5x 2 4x 3 ; c = f(x) = 2x 4 + x 3 14x 2 + 5x + 6 ; c = f(x)= 4x 3 6x 2 + 8x 3; c = ½ 15. f(x) = 27x 4 9x 3 + 3x 2 + 6x + 1 ; c = -1/3 16. Factor f(x) = 9x 3 + 6x 2 3x if you know (x+1) is a factor. 17. Factor f(x) = x 3 2x 2 9x + 18 if you know (x+3) is a factor. 39
41 18. Factor y = x 3 4x 2 3x +18 if you know that (x+2) is a factor. 19. Show that 3 is a zero of multiplicity 2 of the polynomial function P(x)= x 4 + 7x x 2 3x 18 and express P (x) as a product of linear factors. 20. Show that 1 is a zero of multiplicity 4 of the polynomial function f(x)= x 5 + x 4 6x 3 14x 2 11x 3 and express f (x) as a product of linear factors. 21. Find a polynomial function of degree 4 such that both 2 and 3 are zeros of multiplicity Find a polynomial function of degree 5 such that 2 is a zero of multiplicity 3 and 4 is a zero of multiplicity Determine k so that that f(x ) = x 3 + kx 2 kx +10 is divisible by x Find k so that when x 3 x 2 kx + 10 is divided x 3, the remainder is Find k so that when x 3 kx 2 kx +1 is divided by x-2, the remainder =0 26. Determine k so that that f(x ) = 2kx 3 + 2kx - 10 is divisible by x SOLVE x 3 + 4x 2 5x = 0 completely. 40
42 Polynomial Modelling Word Problems When an equation is NOT given: 1. Define your variable(s) 2. If needed draw a picture. 3. Write an equation(s) to solve the problem. 4. State the solution. 5. Explain in words how you found the solution. EXAMPLES: 1. The length of a rectangular pool is 4 yd longer than its width. The area of the pool is 60 yd. What are the dimensions of the pool? (6 x 10 yds) 2. A rectangle has a perimeter of 52 inches. Find the dimensions of the rectangle with maximum area. (13 x 13 in) 3. Find two consecutive negative integers whose product is 240. (-15 & -16) 4. Find two numbers who sum is 20 and whose product is a maximum (10 & 10). 41
43 Polynomial Word Problem Worksheet 1. Find two consecutive positive integers whose product is Find two numbers who difference is 8 and whose product is a minimum. 3. A rectangle has a perimeter of 48 inches. Find the dimensions of the rectangle with maximum area. 4. Find the negative integer whose square is 10 more than 3 times the integer. 5. One side of a rectangular garden is 2 yd less than the other side. If the area of the garden is 63 yd 2, find the dimensions of the garden. ANSWERS: 1) 21 & 22 2) 4 & -4 3) 12 x 12 in 4) -2 5) 7 x 9 yds. 42
44 6. Find two numbers who sum is -12 and whose product is a maximum. 7. Find 2 numbers whose sum is 36 and whose product is a maximum. 8. Find 2 numbers whose difference is 40 and whose product is a minimum. 9. A rectangle has a perimeter of 40 meters. Find the dimensions of the rectangle with the maximum area. 10. Nick has 120 feet of fencing for a kennel. If his house is to be used as one side of the kennel, find the dimensions to maximize the area. ANSWERS: 6) -6 & -6 7) 18 & 18 8) 20 & -20 9) 10 x 10 10) 60 x 30 43
45 Polynomial Functions Review 1. Complete the table below Function Degree End Behavior A f(x) = 3x 5 x 10 Domain and Range B g(x) = x 2 + 5x + 3 C h(x) = 3(x + 2)(x 4) D j(x) = 2x 3 x 2 + 5x 1 2. Evaluate the polynomial f(x) = 3x 5 x 3 + 6x 2 x + 1 for x = 2. Explain what your answer represents. 3. Find the zeros for the function f(x) = x 3 + 3x 2 x 3 4. Show whether -4 is a zero of g(x) = x 3 x 2 14x Use the graph to answer the following questions a) Relative maximum: b) Relative minimum: c) Increasing interval: d) Decreasing interval: e) Domain: f) Range: g) End Behavior: h) Zeros: Find all the zeros 6. f(x) = 2x 3 + 3x 2 39x 20 and 4 is a zero 7. f(x) = x 4 + 3x 2 4 and 1 is a zero 44
46 Divide using long division 8. x 3 3x 2 + 8x 5 (x 1) 9. 4x 3 12x 2 x + 15 (2x 3) 10. Sketch a graph f(x) = 4(x 1) 2 (x 3)(x + 8) 11. Write the function of x 4 shifted 3 units down, 4 units left, a reflection over the x-axis and a horizontal compression by A cement walk of uniform width surrounds a rectangular swimming pool that is 10 m wide and 50 m long. Find the width of the walk if its area is 864 m The number of eggs, f(x), in a female moth is a function of her abdominal width, x, in millimeters, modeled by f(x) = 14x 3 17x 2 16x What is the abdominal width when there are 211 eggs? 14. A pyramid can be formed using equal-size balls. For example, 3 balls can be arranged in a triangle, then a fourth ball placed in the middle on top of them. The function p(n) = 1 n ( n 1)( n 2) gives the number of balls 6 in a pyramid, where n is the number of balls on each side of the bottom layer. (For the pyramid described above, n = 2. For the pyramid in the picture, n = 5.) a. Evaluate p(2), p(3), and p(4). Sketch a picture of the pyramid that goes with each of these values. Check that your function values agree with your pyramid pictures. b. If you had 1000 balls available and you wanted to make the largest possible pyramid using them, what would be the size of the bottom triangle, and how many balls would you use to make the pyramid? How many balls would be left over? 45
47 Sketch the graphs (no graphing calculator) 1. f(x) = ( 1/5)(x + 3)(x + 5)(x 2) 2 2. f(x) = ( 1/6)(x - 3) 2 (x - 1)(x + 2) 2 y y x x a.) Degree b.) x- intercepts c.) y-intercept d.) Degree e.) x- intercepts f.) y-intercept Graphing Calculator Allowed 3. P(x) = (7x 5 + 3x 9 2x + 4) (5x 2 2x + 4) a) Standard Form: b) Degree c) Classify by the # of terms: 4. Find p(3) for p(x) = -4x 4 + 9x x 2 2x Is (x + 2) a factor of p(x) = x 4 + 3x 3 3x 10? 6. Divide using long division (2x 4 5x 3 + 7x 2 + 2x + 4) (2x 3) 46
48 7. Write a polynomial function in standard form that has zeros at 2, -1, and 3 multiplicity 2? 8. Solve 0 = x 3 x 2 11x + 3 given that -3 is a zero. 9. Is -3 a zero of p(x) = 2x4 + 9x3 7x + 10? Why or why not? 10. Is (x + 7) a factor of p(x) = x4 + 9x3 + 15x2 + 5x 14? Why or why not? 11. Find p(3) for p(x) = 3x4 11x3 x2 + 15x 12? 12. Factor p(x) = 3x3 + 14x2 7x 10 completely, given p(-5) = Write the polynomial in factored form with zeros: 1 multiplicity 3, 0, -4? 14. Solve p(x) = x 3 3x 2 11x 7 given that -1 is a zero. 15. Factor p(x) = 6x 3 23x 2 6x + 8 if (x 4) is a factor. 47
49 16. Sketch the graph of p(x) = -1(x 2)(x + 3)(x + 1) (no calc) 17. Solve p(x) = x 3 11x x 36 if (x 6) is a factor. 18. Solve p(x) = 15x 3 119x 2 10x + 16 if 8 is a zero. 19. Divide x4 3x x 2 12x + 16 by x 3 using long division. 20. One root of 2x 3 10x 2 + 9x 4 = 0 is 4. Find the other roots. 21. If 3 + 2i is a zero of a polynomial, what has to be another zero? 22. Approximate to the nearest tenth the real zeros of f(x) = x3 6x2 + 8x 2. (Use a calculator) 23. Write a polynomial function with zeros 1 and 2 (of multiplicity 3) in standard form. 24. Use synthetic division to find f( 2) if f(x) = 4x5 + 10x4 11x3 22x2 + 20x
50 25. Factor: 2x3 + 15x2 14x 48 if (x 2) is a factor. 26. Determine if the degree of the functions below is even or odd. How many real zeros does each have? a) b) c) 49
Dividing Polynomials
5.3 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS 2A.7.C Dividing Polynomials Essential Question How can you use the factors of a cubic polynomial to solve a division problem involving the polynomial? Dividing
More informationUse direct substitution to evaluate the polynomial function for the given value of x
Checkpoint 1 Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient 1. f(x) = 8 x 2 2. f(x) = 6x + 8x 4 3 3. g(x) = πx
More informationChapter 2 Notes: Polynomials and Polynomial Functions
39 Algebra 2 Honors Chapter 2 Notes: Polynomials and Polynomial Functions Section 2.1: Use Properties of Exponents Evaluate each expression (3 4 ) 2 ( 5 8 ) 3 ( 2) 3 ( 2) 9 ( a2 3 ( y 2 ) 5 y 2 y 12 rs
More information24. Find, describe, and correct the error below in determining the sum of the expressions:
SECONDARY 3 HONORS ~ Unit 2A Assignments SECTION 2.2 (page 69): Simplify each expression: 7. 8. 9. 10. 11. Given the binomials and, how would you find the product? 13. Is the product of two polynomials
More informationUsing Properties of Exponents
6.1 Using Properties of Exponents Goals p Use properties of exponents to evaluate and simplify expressions involving powers. p Use exponents and scientific notation to solve real-life problems. VOCABULARY
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area
More informationSecondary Math 3 Honors - Polynomial and Polynomial Functions Test Review
Name: Class: Date: Secondary Math 3 Honors - Polynomial and Polynomial Functions Test Review 1 Write 3x 2 ( 2x 2 5x 3 ) in standard form State whether the function is even, odd, or neither Show your work
More informationGrade 12 Pre-Calculus Mathematics Notebook. Chapter 3. Polynomial Functions
Grade 1 Pre-Calculus Mathematics Notebook Chapter 3 Polynomial Functions Outcomes: R11 & R1 3.1 Characteristics of Polynomial Functions R1 (p.106-113) Polynomial Function = a function of the form where
More informationSubtract 16 from both sides. Divide both sides by 9. b. Will the swing touch the ground? Explain how you know.
REVIEW EXAMPLES 1) Solve 9x + 16 = 0 for x. 9x + 16 = 0 9x = 16 Original equation. Subtract 16 from both sides. 16 x 9 Divide both sides by 9. 16 x Take the square root of both sides. 9 4 x i 3 Evaluate.
More information6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property
6.1 Using Properties of Exponents Objectives 1. Use properties of exponents to evaluate and simplify expressions involving powers. 2. Use exponents and scientific notation to solve real life problems.
More informationSolving Polynomial Equations 3.5. Essential Question How can you determine whether a polynomial equation has a repeated solution?
3. Solving Polynomial Equations Essential Question Essential Question How can you determine whether a polynomial equation has a repeated solution? Cubic Equations and Repeated Solutions USING TOOLS STRATEGICALLY
More informationUP AND UP DOWN AND DOWN DOWN AND UP UP AND DOWN
1. IDENTIFY END BEHAVIOR OF A POLYNOMIAL FROM A GRAPH End behavior is the direction of the graph at the left and the right. There are four options for end behavior: up and up, down and down, down and up,
More information2.1 Quadratic Functions
Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.
More informationTopic 25: Quadratic Functions (Part 1) A quadratic function is a function which can be written as 2. Properties of Quadratic Functions
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 1 of 3 Topic 5: Quadratic Functions (Part 1) Definition: A quadratic function is a function which can be written as f x ax bx
More information2 P a g e. Essential Questions:
NC Math 1 Unit 5 Quadratic Functions Main Concepts Study Guide & Vocabulary Classifying, Adding, & Subtracting Polynomials Multiplying Polynomials Factoring Polynomials Review of Multiplying and Factoring
More informationCharacteristics of Polynomials and their Graphs
Odd Degree Even Unit 5 Higher Order Polynomials Name: Polynomial Vocabulary: Polynomial Characteristics of Polynomials and their Graphs of the polynomial - highest power, determines the total number of
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 informationWhich one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x ) A) x = 5 B) x = -6 C) x = -5 D) x = 6
Review for Final Exam Math 124A (Flatley) Name Which one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x - 14 1) A) x = 5 B) x = -6 C) x = -5 D) x = 6 Solve the linear equation.
More information3.3 Real Zeros of Polynomial Functions
71_00.qxp 12/27/06 1:25 PM Page 276 276 Chapter Polynomial and Rational Functions. Real Zeros of Polynomial Functions Long Division of Polynomials Consider the graph of f x 6x 19x 2 16x 4. Notice in Figure.2
More informationHonors Algebra 2. a.) c.) d.) i and iv only. 3.) How many real roots must the following equation have? a.) 1 b.) 2 c.) 4 d.) none. a.) b.) c.) d.
Honors Algebra 2 The Polynomial Review Name: Date: Period: 1.) What is the remainder when p(x) = x 6 2x 3 + x 1 is divided by (x + 1)? 3 1 1 3 2.) If p(x) = x 3 2x 2 + 9x 2, which of the following statement(s)
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 informationUNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS
UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS This unit investigates quadratic functions. Students study the structure of quadratic expressions and write quadratic expressions in equivalent forms.
More informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic
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 informationChapter 2 Polynomial and Rational Functions
SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear
More informationPolynomial and Synthetic Division
Polynomial and Synthetic Division Polynomial Division Polynomial Division is very similar to long division. Example: 3x 3 5x 3x 10x 1 3 Polynomial Division 3x 1 x 3x 3 3 x 5x 3x x 6x 4 10x 10x 7 3 x 1
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 informationChapter 3-1 Polynomials
Chapter 3 notes: Chapter 3-1 Polynomials Obj: SWBAT identify, evaluate, add, and subtract polynomials A monomial is a number, a variable, or a product of numbers and variables with whole number exponents
More informationAlgebra 32 Midterm Review Packet
Algebra 2 Midterm Review Packet Formula you will receive on the Midterm: x = b ± b2 4ac 2a Name: Teacher: Day/Period: Date of Midterm: 1 Functions: Vocabulary: o Domain (Input) & Range (Output) o Increasing
More informationLet's look at some higher order equations (cubic and quartic) that can also be solved by factoring.
GSE Advanced Algebra Polynomial Functions Polynomial Functions Zeros of Polynomial Function Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring. In the video,
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 informationInt Math 3 Midterm Review Handout (Modules 5-7)
Int Math 3 Midterm Review Handout (Modules 5-7) 1 Graph f(x) = x and g(x) = 1 x 4. Then describe the transformation from the graph of f(x) = x to the graph 2 of g(x) = 1 2 x 4. The transformations are
More informationMATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.
MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)
More informationSection 3.1: Characteristics of Polynomial Functions
Chapter 3: Polynomial Functions Section 3.1: Characteristics of Polynomial Functions pg 107 Polynomial Function: a function of the form f(x) = a n x n + a n 1 x n 1 +a n 2 x n 2 +...+a 2 x 2 +a 1 x+a 0
More information6.5 Dividing Polynomials
Name Class Date 6.5 Dividing Polynomials Essential Question: What are some ways to divide polynomials, and how do you know when the divisor is a factor of the dividend? Explore Evaluating a Polynomial
More informationChapter 2 notes from powerpoints
Chapter 2 notes from powerpoints Synthetic division and basic definitions Sections 1 and 2 Definition of a Polynomial Function: Let n be a nonnegative integer and let a n, a n-1,, a 2, a 1, a 0 be real
More informationPolynomial Functions. 6A Operations with Polynomials. 6B Applying Polynomial. Functions
Polynomial Functions 6A Operations with Polynomials 6-1 Polynomials 6- Multiplying Polynomials 6-3 Dividing Polynomials Lab Explore the Sum and Difference of Two Cubes 6-4 Factoring Polynomials 6B Applying
More informationDividing Polynomials
3-3 3-3 Dividing Polynomials Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Divide using long division. 1. 161 7 2. 12.18 2.1 23 5.8 Divide. 3. 4. 6x + 15y 3 7a 2 ab a 2x + 5y 7a b Objective
More informationPower Functions and Polynomial Functions
CHAPTER Power Functions and Polynomial Functions Estuaries form when rivers and streams meet the sea, resulting in a mix of salt and fresh water. On the coast of Georgia, large estuaries have formed where
More informationSynthetic Substitution
Write your questions and thoughts here! 7.2 Synthetic and Long Polynomial Division 1 Use direct substitution to evaluate t a = a! a! + a + 5 when a = - 2 Synthetic Substitution Synthetic substitution is
More informationRight Behavior. Left Behavior. Right Behavior
U n i t 3 P a r t P a g e 1 Math 3 Unit 3 Part Day 1 Graphing Polynomial Functions Expression 9 x- 3x x + 4x 3 + x + x + 1 5x 4 + x + 10 X 5 + x + 5 3c + 4c /c Type of Function Left Behavior: Right Behavior:
More informationUnit 1: Polynomial Functions SuggestedTime:14 hours
Unit 1: Polynomial Functions SuggestedTime:14 hours (Chapter 3 of the text) Prerequisite Skills Do the following: #1,3,4,5, 6a)c)d)f), 7a)b)c),8a)b), 9 Polynomial Functions A polynomial function is an
More informationEvaluate and Graph Polynomial Functions
5.2 Evaluate and Graph Polynomial Functions Before You evaluated and graphed linear and quadratic functions. Now You will evaluate and graph other polynomial functions. Why? So you can model skateboarding
More informationPolynomial Functions and Models
1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models
More informationPRE-ALGEBRA SUMMARY WHOLE NUMBERS
PRE-ALGEBRA SUMMARY WHOLE NUMBERS Introduction to Whole Numbers and Place Value Digits Digits are the basic symbols of the system 0,,,, 4,, 6, 7, 8, and 9 are digits Place Value The value of a digit in
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 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 informationUsing the Laws of Exponents to Simplify Rational Exponents
6. Explain Radicals and Rational Exponents - Notes Main Ideas/ Questions Essential Question: How do you simplify expressions with rational exponents? Notes/Examples What You Will Learn Evaluate and simplify
More informationa real number, a variable, or a product of a real number and one or more variables with whole number exponents a monomial or the sum of monomials
5-1 Polynomial Functions Objectives A2.A.APR.A.2 (formerly A-APR.A.3) Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function
More informationCh 7 Summary - POLYNOMIAL FUNCTIONS
Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)
More informationLESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationThe Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function
8/1/015 The Graph of a Quadratic Function Quadratic Functions & Models Precalculus.1 The Graph of a Quadratic Function The Graph of a Quadratic Function All parabolas are symmetric with respect to a line
More information( 3) ( ) ( ) ( ) ( ) ( )
81 Instruction: Determining the Possible Rational Roots using the Rational Root Theorem Consider the theorem stated below. Rational Root Theorem: If the rational number b / c, in lowest terms, is a root
More informationMy Math Plan Assessment #1 Study Guide
My Math Plan Assessment #1 Study Guide 1. Find the x-intercept and the y-intercept of the linear equation. 8x y = 4. Use factoring to solve the quadratic equation. x + 9x + 1 = 17. Find the difference.
More informationMATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.
MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)
More informationMath 3 Variable Manipulation Part 3 Polynomials A
Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does
More informationAlgebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions
Algebra III Chapter 2 Note Packet Name Essential Question: Section 2.1: Polynomial Functions Polynomials -Have nonnegative exponents -Variables ONLY in -General Form n ax + a x +... + ax + ax+ a n n 1
More informationPre-Algebra 2. Unit 9. Polynomials Name Period
Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:
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 informationHigher-Degree Polynomial Functions. Polynomials. Polynomials
Higher-Degree Polynomial Functions 1 Polynomials A polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication,
More informationSecondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics
Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together
More informationSolve for the variable by transforming equations:
Cantwell Sacred Heart of Mary High School Math Department Study Guide for the Algebra 1 (or higher) Placement Test Name: Date: School: Solve for the variable by transforming equations: 1. y + 3 = 9. 1
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 informationWhy? _ v a There are different ways to simplify the expression. one fraction. term by 2a. = _ b 2
Dividing Polynomials Then You divided rational expressions. (Lesson 11-5) Now 1Divide a polynomial by a monomial. 2Divide a polynomial by a binomial. Why? The equation below describes the distance d a
More informationPower and Polynomial Functions. College Algebra
Power and Polynomial Functions College Algebra Power Function A power function is a function that can be represented in the form f x = kx % where k and p are real numbers, and k is known as the coefficient.
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 information3.3 Dividing Polynomials. Copyright Cengage Learning. All rights reserved.
3.3 Dividing Polynomials Copyright Cengage Learning. All rights reserved. Objectives Long Division of Polynomials Synthetic Division The Remainder and Factor Theorems 2 Dividing Polynomials In this section
More informationSection 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
More informationPre-Calculus Summer Math Packet 2018 Multiple Choice
Pre-Calculus Summer Math Packet 208 Multiple Choice Page A Complete all work on separate loose-leaf or graph paper. Solve problems without using a calculator. Write the answers to multiple choice questions
More informationPolynomial Functions. Essential Questions. Module Minute. Key Words. CCGPS Advanced Algebra Polynomial Functions
CCGPS Advanced Algebra Polynomial Functions Polynomial Functions Picture yourself riding the space shuttle to the international space station. You will need to calculate your speed so you can make the
More informationPolynomial Degree Leading Coefficient. Sign of Leading Coefficient
Chapter 1 PRE-TEST REVIEW Polynomial Functions MHF4U Jensen Section 1: 1.1 Power Functions 1) State the degree and the leading coefficient of each polynomial Polynomial Degree Leading Coefficient y = 2x
More information4x 2-5x+3. 7x-1 HOMEWORK 1-1
HOMEWORK 1-1 As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around,
More informationR1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member
Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers
More informationName: 6.4 Polynomial Functions. Polynomial in One Variable
Name: 6.4 Polynomial Functions Polynomial Functions: The expression 3r 2 3r + 1 is a in one variable since it only contains variable, r. KEY CONCEPT Polynomial in One Variable Words A polynomial of degree
More informationL1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to
More informationSEE the Big Idea. Quonset Hut (p. 218) Zebra Mussels (p. 203) Ruins of Caesarea (p. 195) Basketball (p. 178) Electric Vehicles (p.
Polnomial Functions.1 Graphing Polnomial Functions. Adding, Subtracting, and Multipling Polnomials.3 Dividing Polnomials. Factoring Polnomials.5 Solving Polnomial Equations. The Fundamental Theorem of
More informationCollege Algebra. George Voutsadakis 1. LSSU Math 111. Lake Superior State University. 1 Mathematics and Computer Science
College Algebra George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 111 George Voutsadakis (LSSU) College Algebra December 2014 1 / 71 Outline 1 Higher Degree
More informationChapter 1- Polynomial Functions
Chapter 1- Polynomial Functions Lesson Package MHF4U Chapter 1 Outline Unit Goal: By the end of this unit, you will be able to identify and describe some key features of polynomial functions, and make
More informationAlgebra II Vocabulary Word Wall Cards
Algebra II Vocabulary Word Wall Cards Mathematics vocabulary word wall cards provide a display of mathematics content words and associated visual cues to assist in vocabulary development. The cards should
More informationSection 1.1. Chapter 1. Quadratics. Parabolas. Example. Example. ( ) = ax 2 + bx + c -2-1
Chapter 1 Quadratic Functions and Factoring Section 1.1 Graph Quadratic Functions in Standard Form Quadratics The polynomial form of a quadratic function is: f x The graph of a quadratic function is a
More informationChapter 4E - Combinations of Functions
Fry Texas A&M University!! Math 150!! Chapter 4E!! Fall 2015! 121 Chapter 4E - Combinations of Functions 1. Let f (x) = 3 x and g(x) = 3+ x a) What is the domain of f (x)? b) What is the domain of g(x)?
More information3 Polynomial and Rational Functions
3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,
More information3.5. Dividing Polynomials. LEARN ABOUT the Math. Selecting a strategy to divide a polynomial by a binomial
3.5 Dividing Polynomials GOAL Use a variety of strategies to determine the quotient when one polynomial is divided by another polynomial. LEARN ABOU the Math Recall that long division can be used to determine
More informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More informationCalifornia 5 th Grade Standards / Excel Math Correlation by Lesson Number
(Activity) L1 L2 L3 Excel Math Objective Recognizing numbers less than a million given in words or place value; recognizing addition and subtraction fact families; subtracting 2 threedigit numbers with
More informationGranite School District Parent Guides Utah Core State Standards for Mathematics Grades K-6
Granite School District Parent Guides Grades K-6 GSD Parents Guide for Kindergarten The addresses Standards for Mathematical Practice and Standards for Mathematical Content. The standards stress not only
More informationAlgebra II Vocabulary Cards
Algebra II Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Complex Numbers Complex Number (examples)
More information32. Use a graphing utility to find the equation of the line of best fit. Write the equation of the line rounded to two decimal places, if necessary.
Pre-Calculus A Final Review Part 2 Calculator Name 31. The price p and the quantity x sold of a certain product obey the demand equation: p = x + 80 where r = xp. What is the revenue to the nearest dollar
More informationChapter 7 Polynomial Functions. Factoring Review. We will talk about 3 Types: ALWAYS FACTOR OUT FIRST! Ex 2: Factor x x + 64
Chapter 7 Polynomial Functions Factoring Review We will talk about 3 Types: 1. 2. 3. ALWAYS FACTOR OUT FIRST! Ex 1: Factor x 2 + 5x + 6 Ex 2: Factor x 2 + 16x + 64 Ex 3: Factor 4x 2 + 6x 18 Ex 4: Factor
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 information= 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives:
Math 65 / Notes & Practice #1 / 20 points / Due. / Name: Home Work Practice: Simplify the following expressions by reducing the fractions: 16 = 4 = 8xy =? = 9 40 32 38x 64 16 Solve the following equations
More information2-2: Evaluate and Graph Polynomial Functions
2-2: Evaluate and Graph Polynomial Functions What is a polynomial? -A monomial or sum of monomials with whole number exponents. Degree of a polynomial: - The highest exponent of the polynomial How do we
More informationL1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to
More informationPreCalculus Practice Midterm
Practice Midterm PreCalculus 1 Name: Period: Date: Answer the following questions. 1. Define function. PreCalculus Practice Midterm 2. Describe the end behavior of any positive odd polynomial function
More informationNAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2
5-1 Operations with Polynomials What You ll Learn Skim the lesson. Predict two things that you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary Evaluate
More informationAlgebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals
Algebra 1 Math Review Packet Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals 2017 Math in the Middle 1. Clear parentheses using the distributive
More informationInstructional Materials for the WCSD Math Common Finals
201-2014 Algebra 2 Semester 1 Instructional Materials for the WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Math Common Final blueprint for
More informationChapter 2 Prerequisite Skills BLM Evaluate Functions 1. Given P(x) = x 4 3x 2 + 5x 11, evaluate.
Chapter Prerequisite Skills BLM 1.. Evaluate Functions 1. Given P(x) = x 4 x + 5x 11, evaluate. a) P( ) b) P() c) P( 1) 1 d) P 4 Simplify Expressions. Expand and simplify. a) (x x x + 4)(x 1) + b) (x +
More informationSec 2.1 Operations with Polynomials Polynomial Classification and Operations
Sec.1 Operations with Polynomials Polynomial Classification and Operations Name Examples Non-Examples Monomial 1. x 4 degree:4 or quartic 1. x 4 (one term). a degree: or quadratic. 5 m. 5 degree:0 or constant.
More informationx 2 + x + x 2 x 3 b. x 7 Factor the GCF from each expression Not all may be possible. 1. Find two numbers that sum to 8 and have a product of 12
Factor the GCF from each expression 4 5 1. 15x 3x. 16x 4 Name: a. b. 4 7 3 6 5 3. 18x y 36x y 4x y 5 4. 3x x 3 x 3 c. d. Not all may be possible. 1. Find two numbers that sum to 8 and have a product of
More information