# 5-3 Polynomial Functions. not in one variable because there are two variables, x. and y

Save this PDF as:

Size: px
Start display at page:

Download "5-3 Polynomial Functions. not in one variable because there are two variables, x. and y"

## Transcription

1 y. 5-3 Polynomial Functions State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why x 6 5x 5 + 4x 2 coefficient of the first term of the polynomial when written in standard form. Degree = 6, leading coefficient = 11 degree = 6, leading coefficient = x 7 5x 3 + 4x 22 coefficient of the first term of the polynomial when written in standard form. Degree = 7, leading coefficient = 10 degree = 7, leading coefficient = x 4 9x 3 + 3x 4y coefficient of the first term of the polynomial when written in standard form. This polynomial is not in one variable because there are two variables, x and y. not in one variable because there are two variables, x and y 4. 8x 5 3x 2 + 4xy 5 coefficient of the first term of the polynomial when written in standard form. This polynomial is not in one variable because there are two variables, x and y. not in one variable because there are two variables, x and y 4. 8x 5 3x 2 + 4xy 5 coefficient of the first term of the polynomial when written in standard form. This polynomial is not in one variable because there are two variables, x and y. not in one variable because there are two variables, x and y Find w(5) and w( 4) for each function. 5. w(x) = 2x 3 + 3x 12 w(5) = 247; w( 4) = w(x) = 2x 4 5x 3 + 3x 2 2x + 8 w(5) = 698; w( 4) = 896 If c(x) = 4x 3 5x and d(x) = 3x 2 + 6x 10, find each value. 7. c(y 3 ) not in one variable because there are two variables, x and y esolutions Manual - Powered by Cognero Page 1 4y 9 5y 6 + 2

2 5-3 Polynomial Functions w(5) = 698; w( 4) = 896 If c(x) = 4x 3 5x and d(x) = 3x 2 + 6x 10, find each value. 7. c(y 3 ) 96b b 2 288b 12 For each graph, describe the end behavior, b. determine whether it represents an odddegree or an even-degree function, and c. state the number of real zeros. 4y 9 5y [d(3z)] 11. As the x-values approach negative infinity, the y- values approach As the x-values positive infinity. 108z 2 72z c(4a) + 2d(3a 5) 1536a 3 426a 2 144a + 82 The end behavior is in opposite directions. b. Since the end behavior is in opposite directions, it is an odd-degree function. c. The graph intersects the x-axis at three points, so there are three real zeros. b. Since the end behavior is in opposite directions, it is an odd-degree function. c. The graph intersects the x-axis at three points, so there are three real zeros c(2b) + 6d(4b 3) 96b b 2 288b 12 For each graph, describe the end behavior, b. determine whether it represents an odddegree or an even-degree function, and c. state the number of real zeros. 12. As the x-values approach negative infinity, the y- values approach As the x-values esolutions Manual - Powered by Cognero Page 2 The end behavior is in the same directions.

3 b. Since the end behavior is in opposite directions, it is an odd-degree function. c. The graph intersects the x-axis at three points, so 5-3 Polynomial there are three Functions real zeros. two variables, x and y. not in one variable because there are two variables, x and y a 7 4a As the x-values approach negative infinity, the y- values approach As the x-values The end behavior is in the same directions. c. The graph intersects the x-axis at zero points, so there are no real zeros. not a polynomial in one variable; with an exponent less than 0. has the variable not a polynomial because there is a negative exponent 15. 8x 5 12x x 3 9 coefficient of the first term of a polynomial written in standard form. The degree = 6, leading coefficient = 12. degree = 6, leading coefficient = 12 c. The graph intersects the x-axis at zero points, so there are no real zeros. CCSS PERSEVERANCE State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why x 6 4x xy not a polynomial in one variable because there are two variables, x and y. not in one variable because there are two variables, x and y a 7 4a 4 + not a polynomial in one variable; with an exponent less than 0. has the variable not a polynomial because there is a negative x 2 + 5x 21x 7 coefficient of the first term of a polynomial written in standard form. The degree = 7, leading coefficient = 21. degree = 7, leading coefficient = x 4x 3 + 3x 2 5x 4 coefficient of the first term of a polynomial written in standard form. The degree = 4, leading coefficient = 5. degree = 4, leading coefficient = b 3 9b + 3b 5 18 coefficient of the first term of a polynomial written in standard form. The degree = 5, leading coefficient = 3. esolutions Manual - Powered by Cognero Page 3

5 5-3 Polynomial Functions p( 6) = 546; p (3) = p (x) = 2x 3 + 6x 2 10x p( 6) = 319; p (3) = p (x) = 2x 4 + x 3 4x 2 p( 6) = 156; p (3) = p (x) = x 4 4x 3 + 3x 2 5x + 24 p( 6) = 2232; p (3) = 153 If c(x) = 2x 2 4x + 3 and d(x) = x 3 + x + 1, find each value. 29. c(3a) p( 6) = 2322; p (3) = p (x) = x 3 + 3x a 2 12a d(2a) 40a a + 5 p( 6) = 319; p (3) = p (x) = 2x 4 + x 3 4x c(b 2 ) 2b 4 4b d(4a 2 ) esolutions Manual - Powered by Cognero Page 5

6 5-3 Polynomial Functions 2b 4 4b d(4a 2 ) 2y 4 8y For each graph, describe the end behavior, b. determine whether it represents an odddegree or an even-degree function, and c. state the number of real zeros. 64a 6 + 4a d(4y 3) 64y y 2 104y c(y 2 1) 2y 4 8y For each graph, describe the end behavior, b. determine whether it represents an odddegree or an even-degree function, and c. state the number of real zeros. 35. As the x-values approach negative infinity, the y- values approach positive infinity. As the x-values positive infinity. The end behavior is in the same direction. b. Since the end behavior is in same direction, it is an even-degree function. c. The graph intersects the x-axis at four points, so there are four real zeros. c. The graph intersects the x-axis at four points, so there are four real zeros. 35. As the x-values approach negative infinity, the y- values approach positive infinity. As the x-values positive infinity. 36. As the x-values approach negative infinity, the y- values approach positive infinity. As the x-values esolutions Manual - Powered by Cognero Page 6 The end behavior is in opposite direction. b. Since the end behavior is in opposite direction, it is

7 c. The graph intersects the x-axis at four points, so 5-3 Polynomial Functions there are four real zeros. b. Since the end behavior is in opposite directions, it is an odd-degree function. c. The graph intersects the x-axis at one point, so there is one real zero. 36. As the x-values approach negative infinity, the y- values approach positive infinity. As the x-values 37. As the x-values approach negative infinity, the y- values approach As the x-values positive infinity. The end behavior is in opposite direction. b. Since the end behavior is in opposite direction, it is an odd-degree function. c. The graph intersects the x-axis at one point, so there is one real zero. b. Since the end behavior is in opposite directions, it is an odd-degree function. c. The graph intersects the x-axis at one point, so there is one real zero. The end behavior is in opposite direction. b. Since the end behavior is in opposite direction, it is an odd-degree function. c. The graph intersects the x-axis at one point, so there is one real zero. b. Since the end behavior is in opposite directions, it is an odd-degree function. c. The graph intersects the x-axis at one point, so there is one real zero. 37. As the x-values approach negative infinity, the y- values approach As the x-values positive infinity. 38. As the x-values approach negative infinity, the y- values approach positive infinity. As the x-values positive infinity. The end behavior is in opposite direction. b. Since the end behavior is in opposite direction, it is an odd-degree function. c. The graph intersects the x-axis at one point, so there is one real zero. The end behavior is in same directions. c. The graph intersects the x-axis at zero points, so there are no real zeros. esolutions Manual - Powered by Cognero Page 7

8 b. Since the end behavior is in opposite directions, it is an odd-degree function. 5-3 Polynomial c. The graph Functions intersects the x-axis at one point, so there is one real zero. c. The graph intersects the x-axis at no points, so there are no real zeros. 38. As the x-values approach negative infinity, the y- values approach positive infinity. As the x-values positive infinity. 39. As the x-values approach negative infinity, the y- values approach As the x-values The end behavior is in same directions. c. The graph intersects the x-axis at zero points, so there are no real zeros. c. The graph intersects the x-axis at no points, so there are no real zeros. The end behavior is in the same direction. c. The graph intersects the x-axis at two points, so there are two real zeros. c. The graph intersects the x-axis at two points, so there are two real zeros. 39. As the x-values approach negative infinity, the y- values approach As the x-values 40. As the x-values approach negative infinity, the y- values approach positive infinity. As the x-values The end behavior is in the same direction. c. The graph intersects the x-axis at two points, so there are two real zeros. The end behavior is in the same direction. c. The graph intersects the x-axis at two points, so there are two real zeros. esolutions Manual - Powered by Cognero Page 8

9 5-3 Polynomial c. The graph Functions intersects the x-axis at two points, so there are two real zeros. 40. As the x-values approach negative infinity, the y- values approach positive infinity. As the x-values 42. CCSS MODELING A microwave manufacturing firm has determined that their profit function is P(x) = x x 2 + 6x 355 where x is the number of microwaves sold annually. Graph the profit function using a calculator. b. Determine a reasonable viewing window for the function. c. Approximate all of the zeros of the function using the CALC menu. 41, 27, d. What must be the range of microwaves sold in order for the firm to have a profit? The end behavior is in the same direction. c. The graph intersects the x-axis at two points, so there are two real zeros. c. The graph intersects the x-axis at two points, so there are two real zeros. 41. PHYSICS For a moving object with mass m in kilograms, the kinetic energy KE in joules is given by the function KE(v) = 0.5mv 2, where v represents the speed of the object in meters per second. Find the kinetic energy of an all-terrain vehicle with a mass of 171 kilograms moving at a speed of 11 meters/second. b. Sample answer: [ 500, 500] scl:50 by [ 10,000, 10,000] scl:1000 c. 41.0, 27.1, d. between 28 and 228 microwaves 10,345.5 joules 42. CCSS MODELING A microwave manufacturing firm has determined that their profit function is P(x) = x x 2 + 6x 355 b. Sample answer: [ 500, 500] scl:50 by [ 10,000, 10,000] scl:1000 c. 41.0, 27.1, d. between 28 and 228 microwaves Find p ( 2) and p (8) for each function. 43. p (x) = x 4 + where x is the number of microwaves sold annually. Graph the profit function using a calculator. b. Determine a reasonable viewing window for the function. esolutions Manual - Powered by Cognero Page 9

10 b. Sample answer: [ 500, 500] scl:50 by [ 10,000, 10,000] scl: Polynomial c. 41.0, 27.1, Functions d. between 28 and 228 microwaves Find p ( 2) and p (8) for each function. 43. p (x) = x p(-2) = 28; p (8) = p( 2) = 16; p (8) = p( 2) = 0.5; p (8) = p(-2) = 28; p (8) = 178 p( 2) = 1.5; p (8) = 304 Use the degree and end behavior to match each polynomial to its graph. A. B. p( 2) = 0.5; p (8) = 3112 esolutions Manual - Powered by Cognero Page C.

11 coefficient of the graph is positive. Therefore, the correct choice is D. 5-3 Polynomial Functions p( 2) = 1.5; p (8) = 304 Use the degree and end behavior to match each polynomial to its graph. A. B. C. D. D 48. f (x) = 2x 2 + 8x + 5 Since the degree of the polynomial is 2, the graph must have 1 turning point. And the function is an even degree function. Therefore, the end behavior of the graph must be in the same direction. The correct choice is B. B 49. f (x) = x 4 3x 2 + 6x Since the degree of the polynomial is 4, the graph must have 3 turning points. And the function is an even degree function. Therefore, the end behavior of the graph must be in the same direction. The correct choice is A. A 50. f (x) = 4x 3 4x Since the degree of the polynomial is 3, the graph must have 2 turning point2. And the leading coefficient of the graph is negative. Therefore, the graph opens down. The correct choice is B. C 47. f(x) = x 3 + 3x 2 4x Since the degree of the polynomial is 3, the graph must have 2 turning points. And the leading coefficient of the graph is positive. Therefore, the correct choice is D. D If c(x) = x 3 2x and d(x) = 4x 2 6x + 8, find each value c(a 4) + 3d(a + 5) 3a 3 24a a f (x) = 2x 2 + 8x + 5 Since the degree of the polynomial is 2, the graph must have 1 turning point. And the function is an even degree function. Therefore, the end behavior of 52. 2d(2a + 3) 4c(a 2 + 1) esolutions Manual - Powered by Cognero Page 11

12 5-3 Polynomial 3a 3 24a a Functions d(2a + 3) 4c(a 2 + 1) in order to be profitable? d. Explain why only two of the zeros are considered in part c. Table: 53. 5c(a 2 ) 8d(6 3a) Plot the points on the coordinate plane. And connect them by a smooth curve. 5a 6 298a a d(a 3 ) + 6c(a 4 + 1) 6a a 8 28a 6 + 6a a BUSINESS A clothing manufacturer s profitability can be modeled by p (x) = x x 2 144, where x is the number of items sold in thousands and p (x) is the company s profit in thousands of dollars. Use a table of values to sketch the function. b. Determine the zeros of the function. c. Between what two values should the company sell in order to be profitable? d. Explain why only two of the zeros are considered in part c. Table: b. 6, 2, 2, 6 c. The graph lies in the Quadrant I (both x and y values are positive) between x = 2 and x = 6. Therefore, the company should sell 2000 to 6000 items. d. Sample answer: The negative values should not be considered because the company will not produce negative items. esolutions Manual - Powered by Cognero Page 12

13 d. Sample answer: The negative values should not be considered because the company will not produce negative items. 5-3 Polynomial Functions x-intercepts: x = 2, 1, 3, 4. y-intercept: y = 24. End behavior: b. c. d. b. 6, 2, 2, 6 c and 6000 items d. Sample answer: The negative values should not be considered because the company will not produce negative items. 56. MULTIPLE REPRESENTATIONS Consider g (x) = (x 2)(x + 1)(x 3)(x + 4) ANALYTICAL Determine the x- and y- intercepts, roots, degree, and end behavior of g(x). b. ALGEBRAIC Write the function in standard form. c. TABULAR Make a table of values for the function. d. GRAPHICAL Sketch a graph of the function by plotting points and connecting them with a smooth curve. Degree: 4; degree: 4; x-intercepts: 2, 1, 3, 4; y-intercept: 24; roots: 2, 1, 3, 4, end behavior: as b. g(x) = x 4 5x x + 24 c. x-intercepts: x = 2, 1, 3, 4. y-intercept: y = 24. End behavior: esolutions Manual - Powered by Cognero Page 13 b. d.

14 5-3 Polynomial Functions d. 60. f (x) = 6x 7x 2 As the x-values approach negative infinity, the y- values approach As the x-values Describe the end behavior of the graph of each function. 57. f (x) = 5x 4 + 3x 2 As the x-values approach negative infinity, the y- values approach As the x-values 61. g(x) = 8x 4 + 5x h(x) = 9x 6 5x 7 + 3x g(x) = 2x 5 + 6x 4 As the x-values approach negative infinity, the y- values approach As the x-values positive infinity. 63. CCSS CRITIQUE Shenequa and Virginia are determining the number of zeros of the graph. Is either of them correct? Explain your reasoning. 59. h(x) = 4x 7 + 8x 6 4x As the x-values approach negative infinity, the y- values approach positive infinity. As the x-values 60. f (x) = 6x 7x 2 As the x-values approach negative infinity, the y- values approach As the x-values esolutions Manual - Powered by Cognero Page 14

16 will become a 2-degree function with a positive 5-3 Polynomial Functions leading coefficient. 66. OPEN ENDED Sketch the graph of an even-degree polynomial with 8 real roots, one of them a double root Sample answer: Since even degree functions have an even number of zeros, draw a graph that crosses the x-axis in three places to the left of the origin and three places to the right of the origin. Place the double root at the origin so the function has 8 real roots including a double root. 67. REASONING Determine whether the following statement is always, sometimes, or never true. Explain. A polynomial function that has four real roots is a fourth-degree polynomial. Sometimes; a polynomial function with four real roots may be a sixth-degree polynomial function with two imaginary roots. A polynomial function that has four real roots is at least a fourth-degree polynomial. Sometimes; a polynomial function with four real roots may be a sixth-degree polynomial function with two imaginary roots. A polynomial function that has four real roots is at least a fourth-degree polynomial. 68. WRITING IN MATH Describe what the end behavior of a polynomial function is and how to determine it. Sample answer: The end behavior of a polynomial function is what the graph does as the input value approaches negative and positive infinity. It can be determined by the leading coefficient and the degree of the polynomial. 67. REASONING Determine whether the following statement is always, sometimes, or never true. Explain. A polynomial function that has four real roots is a fourth-degree polynomial. Sometimes; a polynomial function with four real roots may be a sixth-degree polynomial function with two imaginary roots. A polynomial function that has four real roots is at least a fourth-degree polynomial. Sometimes; a polynomial function with four real roots may be a sixth-degree polynomial function with two imaginary roots. A polynomial function that has four real roots is at least a fourth-degree polynomial. Sample answer: The end behavior of a polynomial function is what the graph does as the input value approaches negative and positive infinity. It can be determined by the leading coefficient and the degree of the polynomial. 69. SHORT RESPONSE Four students solved the same math problem. Each student s work is shown below. Who is correct? Student A Student B Student C Student D esolutions Manual - Powered by Cognero Page 16

17 Sample answer: The end behavior of a polynomial function is what the graph does as the input value approaches negative and positive infinity. It can be 5-3 Polynomial determined by Functions the leading coefficient and the degree of the polynomial. 69. SHORT RESPONSE Four students solved the same math problem. Each student s work is shown below. Who is correct? Student A Student B Student C Student D B 71. EXTENDED RESPONSE A company manufactures tables and chairs. It costs \$40 to make each table and \$25 to make each chair. There is \$1440 available to spend on manufacturing each week. Let t = the number of tables produced and c = the number of chairs produced. The manufacturing equation is 40t + 25c =1500. Construct a graph of this equation. b. The company always produces two chairs with each table. Write and graph an equation to represent this situation on the same graph as the one in part c. Determine the number of tables and chairs that the company can produce each week. d. Explain how to determine this answer using the graph. Graph the equation. Student A is correct. Student A 70. SAT/ACT What is the remainder when x 3 7x + 5 is divided by x + 3? A 11 B 1 C 1 D 11 E 35 Use synthetic division to divide. b. Since the company produces two chairs with each table, the equation should be 2t = c. That is, t = 0.5c. Graph this equation on the same coordinate plane. The remainder is 1. So, the correct choice is B. B 71. EXTENDED RESPONSE A company manufactures tables and chairs. It costs \$40 to make each table and \$25 to make each chair. There is \$1440 available to spend on manufacturing each week. Let t = the number of tables produced and c = the number of chairs produced. c. The graphs are intersecting at the point (32, 16). Therefore, 32 chairs and 16 tables can be produced each week. d. Sample answer: This can be determined by the intersection of the graphs. This point of intersection is esolutions Manual - Powered by Cognero Page 17

18 5-3 Polynomial Functions c. The graphs are intersecting at the point (32, 16). Therefore, 32 chairs and 16 tables can be produced each week. d. Sample answer: This can be determined by the intersection of the graphs. This point of intersection is the optimal amount of tables and chairs manufactured. c. 16 tables and 32 chairs d. Sample answer: This can be determined by the intersection of he graphs. This point of intersection is the optimal amount of tables and chairs manufactured. 72. If i = then 5i(7i) = F 70 H 35 G 35 J 70 The correct choice is H. H Simplify. b. t = 0.5c; 73. 2x 2 y 2 + 4x 4 y 4 z c. 16 tables and 32 chairs d. Sample answer: This can be determined by the intersection of he graphs. This point of intersection is the optimal amount of tables and chairs manufactured. 72. If i = then 5i(7i) = F 70 H 35 G 35 J b 3 c 3 5a 3 b 2 + 2a 4 c esolutions Manual - Powered by Cognero Page 18

19 5-3 Polynomial Functions 3b 3 c 3 5a 3 b 2 + 2a 4 c 75. 6c a 5 cd 2 Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial x 2 + 5xy 3 6x + 4 Yes. A polynomial is an expression consisting of monomials in which the exponents are nonnegative integers. The degree of a polynomial is the degree of the monomial with the greatest degree. The monomial with the greatest degree is degree of the polynomial is 4.. The This is not a polynomial since one term has a variable with the exponent less than 0. not a polynomial 79. FOUNTAINS The height of a fountain s water stream can be modeled by a quadratic function. Suppose the water from a jet reaches a maximum height of 8 feet at a distance 1 foot away from the jet. If the water lands 3 feet away from the jet, find a quadratic function that models the height h(d) of the water at any given distance d feet from the jet. Then compare the graph of the function to the parent function. b. Suppose a worker increases the water pressure so that the stream reaches a maximum height of 12.5 feet at a distance of 15 inches from the jet. The water now lands 3.75 feet from the jet. Write a new quadratic function for h(d). How do the changes in h and k affect the shape of the graph? yes; x x 6 16 Yes. A polynomial is an expression consisting of monomials in which the exponents are nonnegative integers. The degree of a polynomial is the degree of the monomial with the greatest degree. The monomial with the greatest degree is degree of the polynomial is 6.. The Find the vertex of the parabol yes; x 4 + 2x 2 x 1 This is not a polynomial since one term has a variable with the exponent less than 0. not a polynomial 79. FOUNTAINS The height of a fountain s water stream can be modeled by a quadratic function. Suppose the water from a jet reaches a maximum From the graph the vertex of the parabola is (1, 8). Therefore, use the vertex form of the equation of the parabol esolutions Manual - Powered by Cognero Page 19

20 The minimum value of the function is From the graph the vertex of the parabola is (1, 8). 5-3 Polynomial Therefore, use Functions the vertex form of the equation of the parabol 81. The graph passes through the point (3, 0). So, substitute the point (3, 0) for (d, h(d)) in the equation to find Therefore, the function is: The graph opens downward and is narrower than the parent graph, and the vertex is at (1, 8). b. Here, the vertex of the new quadratic function is (15 in., 12.5 ft). That is (1.25ft, 12.5ft). The equation of the new quadratic function is: 82. Since the absolute value can not be less than a negative value, the inequality has no solution. It shifted the graph up 4.5 ft and to the right 3 in. h(d) = 2d 2 + 4d + 6; The graph opens downward and is narrower than the parent graph, and the vertex is at (1, 8). b. h(d) = 2(d 1.25) ; It shifted the graph up 4.5 ft and to the right 3 in. Solve each inequality. 80. no solution Determine whether each function has a maximum or minimum value, and find that value. 83. f (x) = 3x 2 8x + 4 For this function a = 3 > 0. Therefore, the graph opens up and the function has a minimum value. The minimum value of the function is the y- coordinate of the vertex. The x-coordinate of the vertex is: 81. The y-coordinate of the vertex is: esolutions Manual - Powered by Cognero Page 20

21 Since the absolute value can not be less than a negative value, the inequality has no solution. 5-3 Polynomial Functions no solution Determine whether each function has a maximum or minimum value, and find that value. 83. f (x) = 3x 2 8x + 4 For this function a = 3 > 0. Therefore, the graph opens up and the function has a minimum value. The minimum value of the function is the y- coordinate of the vertex. The x-coordinate of the vertex is: minimum; 84. f (x) = 4x 2 + 2x 10 For this function a = 4 < 0. Therefore, the graph opens down and the function has a maximum value. The maximum value of the function is the y- coordinate of the vertex. The x-coordinate of the vertex is: The y-coordinate of the vertex is: The y-coordinate of the vertex is: The maximum value of the function is maximum; 9.75 The minimum value of the function is minimum; 84. f (x) = 4x 2 + 2x 10 For this function a = 4 < 0. Therefore, the graph opens down and the function has a maximum value. The maximum value of the function is the y- coordinate of the vertex. The x-coordinate of the vertex is: 85. f (x) = 0.25x 2 + 4x 5 For this function a = 0.25 < 0. Therefore, the graph opens down and the function has a maximum value. The maximum value of the function is the y- coordinate of the vertex. The x-coordinate of the vertex is: The y-coordinate of the vertex is: The maximum value of the function is 11. The y-coordinate of the vertex is: maximum;11 The maximum value of the function is esolutions Manual - Powered by Cognero Page 21

### ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### 2-5 Rational Functions

-5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4

Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

### 8 Polynomials Worksheet

8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph

### 2-2 Linear Relations and Functions. So the function is linear. State whether each function is a linear function. Write yes or no. Explain.

1. 2. 3. 4. State whether each function is a linear function. Write yes or no. Explain. The function written as. is linear as it can be + b. cannot be written in the form f (x) = mx So the function is

### 7.1 Graphs of Quadratic Functions in Vertex Form

7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called

### M 1310 4.1 Polynomial Functions 1

M 1310 4.1 Polynomial Functions 1 Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a, a,..., a, a, a n n1 2 1 0, be real numbers, with a

### is the degree of the polynomial and is the leading coefficient.

Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs

### 8-8 Differences of Squares. Factor each polynomial. 1. x 9 SOLUTION: 2. 4a 25 SOLUTION: 3. 9m 144 SOLUTION: 4. 2p 162p SOLUTION: 5.

Factor each polynomial. 1.x 9 SOLUTION:.a 5 SOLUTION:.9m 1 SOLUTION:.p 16p SOLUTION: 5.u 81 SOLUTION: Page 1 5.u 81 SOLUTION: 6.d f SOLUTION: 7.0r 5n SOLUTION: 8.56n c SOLUTION: Page 8.56n c SOLUTION:

### 6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### Zeros of Polynomial Functions

Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

### Section 3.1 Quadratic Functions and Models

Section 3.1 Quadratic Functions and Models DEFINITION: A quadratic function is a function f of the form fx) = ax 2 +bx+c where a,b, and c are real numbers and a 0. Graphing Quadratic Functions Using the

### a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F

FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

### 1 Functions, Graphs and Limits

1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its

### This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.

COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Section 3.2 Polynomial Functions and Their Graphs

Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P(x) = 3, Q(x) = 4x 7, R(x) = x 2 +x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 +2x+4 (b)

### Algebra Cheat Sheets

Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

### The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying

### MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial

### Week 1: Functions and Equations

Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter

### Polynomial Expressions and Equations

Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

### 2.5 Transformations of Functions

2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### 7-2 Solving Exponential Equations and Inequalities. Solve each equation. 1. 3 5x = 27 2x 4 SOLUTION:

7-2 Solving Exponential Equations and Inequalities Solve each equation. 1. 3 5x = 27 2x 4 3. 2 6x = 32 x 2 12 2. 16 2y 3 = 4 y + 1 10 4. 49 x + 5 = 7 8x 6 3. 2 6x = 32 x 2 5. SCIENCE Mitosis is a process

### 1.1 Practice Worksheet

Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Factoring Polynomials

UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

### Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

### FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

### BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES.

BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES. I. GENERALITIES There are 3 common methods to solve quadratic inequalities. Therefore, students sometimes are confused to select the fastest and the best

74 In the Herb Business, Part III Factoring and Quadratic Equations In the herbal medicine business, you and your partner sold 120 bottles of your best herbal medicine each week when you sold at your original

### CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

### PARABOLAS AND THEIR FEATURES

STANDARD FORM PARABOLAS AND THEIR FEATURES If a! 0, the equation y = ax 2 + bx + c is the standard form of a quadratic function and its graph is a parabola. If a > 0, the parabola opens upward and the

### Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

### 8-5 Using the Distributive Property. Use the Distributive Property to factor each polynomial. 1. 21b 15a SOLUTION:

Use the Distributive Property to factor each polynomial. 1. 1b 15a The greatest common factor in each term is 3.. 14c + c The greatest common factor in each term is c. 3. 10g h + 9gh g h The greatest common

### Some Lecture Notes and In-Class Examples for Pre-Calculus:

Some Lecture Notes and In-Class Examples for Pre-Calculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax

### Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -

### 1.3. Maximum or Minimum of a Quadratic Function. Investigate A

< P1-6 photo of a large arched bridge, similar to the one on page 292 or p 360-361of the fish book> Maximum or Minimum of a Quadratic Function 1.3 Some bridge arches are defined by quadratic functions.

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x- value and L be the y-values for a graph. 1. How are the x and y-values related? What pattern do you see? To enter the

### COMPETENCY TEST SAMPLE TEST. A scientific, non-graphing calculator is required for this test. C = pd or. A = pr 2. A = 1 2 bh

BASIC MATHEMATICS COMPETENCY TEST SAMPLE TEST 2004 A scientific, non-graphing calculator is required for this test. The following formulas may be used on this test: Circumference of a circle: C = pd or

### WARM UP EXERCSE. 2-1 Polynomials and Rational Functions

WARM UP EXERCSE Roots, zeros, and x-intercepts. x 2! 25 x 2 + 25 x 3! 25x polynomial, f (a) = 0! (x - a)g(x) 1 2-1 Polynomials and Rational Functions Students will learn about: Polynomial functions Behavior

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

### Practice Test Answer and Alignment Document Mathematics: Algebra II Performance Based Assessment - Paper

The following pages include the answer key for all machine-scored items, followed by the rubrics for the hand-scored items. - The rubrics show sample student responses. Other valid methods for solving

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### 6.2 Solving Nonlinear Equations

6.2. SOLVING NONLINEAR EQUATIONS 399 6.2 Solving Nonlinear Equations We begin by introducing a property that will be used extensively in this and future sections. The zero product property. If the product

### 1.3 Algebraic Expressions

1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

### Polynomials and Factoring

Lesson 2 Polynomials and Factoring A polynomial function is a power function or the sum of two or more power functions, each of which has a nonnegative integer power. Because polynomial functions are built

### Warm-Up Oct. 22. Daily Agenda:

Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment

### Algebra II A Final Exam

Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

### DRAFT. Algebra 1 EOC Item Specifications

DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as

### Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### Procedure for Graphing Polynomial Functions

Procedure for Graphing Polynomial Functions P(x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine

### Algebra II Unit Number 4

Title Polynomial Functions, Expressions, and Equations Big Ideas/Enduring Understandings Applying the processes of solving equations and simplifying expressions to problems with variables of varying degrees.

### Zeros of Polynomial Functions

Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

### Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### Graphic Designing with Transformed Functions

Math Objectives Students will be able to identify a restricted domain interval and use function translations and dilations to choose and position a portion of the graph accurately in the plane to match

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### CONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form <==> Intercept Form <==> Vertex Form) (By Nghi H Nguyen Dec 08, 2014)

CONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form Intercept Form Vertex Form) (By Nghi H Nguyen Dec 08, 2014) 1. THE QUADRATIC FUNCTION IN INTERCEPT FORM The graph of the quadratic

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Student Name: Teacher: Date: District: Miami-Dade County Public Schools Assessment: 9_12 Mathematics Algebra II Interim 2. Mid-Year 2014 - Algebra II

Student Name: Teacher: District: Date: Miami-Dade County Public Schools Assessment: 9_12 Mathematics Algebra II Interim 2 Description: Mid-Year 2014 - Algebra II Form: 201 1. During a physics experiment,

### Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

### 10 7, 8. 2. 6x + 30x + 36 SOLUTION: 8-9 Perfect Squares. The first term is not a perfect square. So, 6x + 30x + 36 is not a perfect square trinomial.

Squares Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1.5x + 60x + 36 SOLUTION: The first term is a perfect square. 5x = (5x) The last term is a perfect

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### 3.3 Real Zeros of Polynomials

3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

### 63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.

9.4 (9-27) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27-in. wheel, 44 teeth

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF

Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials

### Lyman Memorial High School. Pre-Calculus Prerequisite Packet. Name:

Lyman Memorial High School Pre-Calculus Prerequisite Packet Name: Dear Pre-Calculus Students, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These

### MATH 21. College Algebra 1 Lecture Notes

MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a