Given a polynomial and one of its factors, find the remaining factors of the polynomial. 4. x 3 6x x 6; x 1 SOLUTION: Divide by x 1.
|
|
|
- Heather Ross
- 6 years ago
- Views:
Transcription
1 Use synthetic substitution to find f (4) and f ( 2) for each function. 2. f (x) = x 4 + 8x 3 + x 2 4x 10 Divide the function by x 4. The remainder is 758. Therefore, f (4) = 758. Divide the function by x + 2. The remainder is 46. Therefore, f ( 2) = 46. Given a polynomial and one of its factors, find the remaining factors of the polynomial. 4. x 3 6x x 6; x 1 Divide by x 1. So, the remaining factors of the polynomial are x 3 and x 2. esolutions Manual - Powered by Cognero Page 1
2 6. 3x x 2 x 12; x 1 Divide by x 1. So, the remaining factors of the polynomial are x + 3 and 3x + 4. Use synthetic substitution to find f ( 5) and f (2) for each function. 8. f (x) = x 3 + 2x 2 3x + 1 Divide the function by x + 5. The remainder is 59. Therefore, f ( 5) = 59. Divide the function by x 2. The remainder is 11. Therefore, f (2) = 11. esolutions Manual - Powered by Cognero Page 2
3 10. f (x) = 3x 4 + x 3 2x 2 + x + 12 Divide the function by x + 5. The remainder is Therefore, f ( 5) = Divide the function by x 2. The remainder is 62. Therefore, f (2) = f (x) = x 3 5x + 2 Divide the function by x + 5. The remainder is 98. Therefore, f ( 5) = 98. Divide the function by x 2. The remainder is 0. Therefore, f (2) = 0. esolutions Manual - Powered by Cognero Page 3
4 14. f (x) = x 6 4x 4 + 3x 2 10 Divide the function by x + 5. The remainder is Therefore, f ( 5) = 13,190. Divide the function by x 2. The remainder is 2. Therefore, f (2) = FINANCIAL LITERACY A specific car s fuel economy in miles per gallon can be approximated by f (x) = x x x x 0.38, where x represents the car s speed in miles per hour. Determine the fuel economy when the car is travelling 40, 50 and 60 miles per hour. Find f (40). Find f (50). Find f (60). The fuel economy when the car is travelling 40, 50 and 60 miles per hour are about mpg, mpg and mpg respectively. esolutions Manual - Powered by Cognero Page 4
5 Given a polynomial and one of its factors, find the remaining factors of the polynomial. 18. x 4 + 2x 3 8x 16; x + 2 Divide by x + 2. So, the remaining factors of the polynomial are x 2 and x 2 + 2x x 3 x 2 5x 3; x 3 Divide by x + 2. So, the remaining factor of the polynomial is (x + 1) 2. esolutions Manual - Powered by Cognero Page 5
6 22. 2x 3 + 7x 2 53x 28; x 4 Divide by x 4. So, the remaining factors of the polynomial are x + 7, 2x x 3 + 2x 2 x 2; x + 2 Divide by x + 2. So, the remaining factors of the polynomial are x + 1 and x x 5 32x 4 81x + 162; 2x 3 Divide by 2x 3. So, the remaining factors of the polynomial are x 2, 2x + 3 and 4x esolutions Manual - Powered by Cognero Page 6
7 28. CCSS REASONING A company s sales, in millions of dollars, of consumer electronics can be modeled by S(x) = 1.2x x x + 678, where x is the number of years since a. Use synthetic substitution to estimate the sales for 2017 and b. Do you think this model is useful in estimating future sales? Explain. a. Find S(12) for sales for 2017: Find S(15) for sales for 2020: The sales for 2017 and 2020 are $ million and $1074 million respectively. b. Sample answer: The graph of the function has a relative maximum at about x = 11 or the year 2016 and then the values rapidly decrease. This model could be useful for the next 15 years. After that, is is unlikely that the sales would decrease as rapidly as indicated. esolutions Manual - Powered by Cognero Page 7
8 Use the graphs to find all of the factors for each polynomial function. 30. The depressed polynomial is. The factors of the polynomial function are x 2, 4x 3, and 5x + 2. CCSS Find values of k so that each remainder is (x 2 x + k) (x 1) Divide using synthetic division. So, the value of k is 3. esolutions Manual - Powered by Cognero Page 8
9 34. (x 2 + 5x + 7) (x + k) Divide using synthetic division. Thus, k = 1 and k = OPEN ENDED Write a polynomial function that has a double root of 1 and a double root of 5. Graph the function. Sample answer: A polynomial function with 2 double roots is at least a 4th degree function. Even degree functions have end behavior in the same direction. Since the double roots are 1 and -5, the function will have (x - 1) 2 and (x + 5) 2 as factors. f(x) = 0.1(x 1) 2 (x + 5) 2 esolutions Manual - Powered by Cognero Page 9
10 CHALLENGE Find the solutions of each polynomial function. 38. (x 2 + 3) 2 7(x 2 + 3) + 12 = 0 Substitute x for u. 40. CHALLENGE Review the definition for the Factor Theorem. Provide a proof of the theorem. If x a is a factor of f (x), then f (a) has a factor of (a a) or 0. Since a factor of f (a) is 0, f (a) = 0. Now assume that f (a) = 0. If f (a) = 0, then the Remainder Theorem states that the remainder is 0 when f (x) is divided by x a. This means that x a is a factor of f (x). This proves the Factor Theorem. 42. CHALLENGE Show that the quadratic function ax 4 + bx 3 + cx 2 + dx + e = 0 will always have a rational root when the numbers 1, 2, 3, 4, and 6 are randomly assigned to replace a through f, and all of the numbers are used. Sample answer: When x = 1, f (1) is the sum of all of the coefficients and constants in f (x), in this case, a, b, c, d, and e. The sum of a, b, c, d, and e is 0, so however the coefficients are arranged, f (1) will always equal 0, and f (x) will have a rational root x 3 + y 3 = A (3x + y)(3x + y)(3x + y) B (3x + y)(9x 2 3xy + y 2 ) C (3x y)(9x 2 + 3xy + y 2 ) D (3x y)(9x 2 + 9xy + y 2 ) So, B is the correct option. esolutions Manual - Powered by Cognero Page 10
11 46. What is the product of the complex numbers (4 + i)(4 i)? F 15 G 16 i H 17 J 17 8i H is the correct option. Solve each equation. 48. x 4 4x 2 21 = 0 Substitute x 2 for u. esolutions Manual - Powered by Cognero Page 11
12 50. 4x 4 8x 2 96 = 0 Substitute 2x 2 for u. Complete each of the following. a. Estimate the x coordinate of every turning point and determine if those coordinates are relative maxima or relative minima. b. Estimate the x-coordinate of every zero. c. Determine the smallest possible degree of the function. d. Determine the domain and range of the function. 52. a. The value of f (x) at about x = 2 is greater than the surrounding points, so there must be a relative maximum occur near x = 2. The value of f (x) at about x = 3 is greater than the surrounding points, so there must be a relative maximum occur near x = 3. The value of f (x) at about x = 0 is less than the surrounding points, so there must be a relative minimum near x = 0. The value of f (x) at about x = 3.5 is less than the surrounding points, so there must be a relative minimum near x = 3.5. b. The graph intersects the x-axis at about 3.75, 1, 1.5, 3.25 and Therefore, the zeros of the function are at 3.75, 1, 1.5, 3.25 and c. Since the graph has 4 turning points, the smallest possible degree of the function is 5. d. esolutions Manual - Powered by Cognero Page 12
13 54. HIGHWAY SAFETY Engineers can use the formula d = 0.05v v to estimate the minimum stopping distance d in feet for a vehicle traveling v miles per hour. If a car is able to stop after 125 feet, what is the fastest it could have been traveling when the driver first applied the brakes? Substitute 125 for d in the formula and then use the Quadratic Formula to solve for v. The car could travel about 40.2 mph, when the driver first applied the brakes. Solve by graphing x + 2y = 8 4x + 6y = 11 esolutions Manual - Powered by Cognero Page 13
14 If c(x) = x 2 2x and d(x) = 3x 2 6x + 4, find each value. 58. c(a + 2) d(a 4) Substitute a 2 + 2a for c(a + 2) and for d(a 4) in the expression. 60. c( 3a) + d(a + 4) Substitute 9a 2 + 6a for c( 3a) and for d(a + 4) in the expression. esolutions Manual - Powered by Cognero Page 14
15 62. c(a) + 5d(2a) Substitute a 2 2a for c(a) and for d(2a) in the expression. esolutions Manual - Powered by Cognero Page 15
5-6 The Remainder and Factor Theorems
Use synthetic substitution to find f (4) and f ( 2) for each function. 1. f (x) = 2x 3 5x 2 x + 14 58; 20 2. f (x) = x 4 + 8x 3 + x 2 4x 10 758; 46 3. NATURE The approximate number of bald eagle nesting
2-4 Zeros of Polynomial Functions
Write a polynomial function of least degree with real coefficients in standard form that has the given zeros. 33. 2, 4, 3, 5 Using the Linear Factorization Theorem and the zeros 2, 4, 3, and 5, write f
Plot the points on the coordinate plane and connect them by a smooth curve.
Graph each polynomial equation by making a table of values. 2. f (x) = 2x 4 + 4x 3 + 2x 2 + x 3 Make a table of values. Plot the points on the coordinate plane and connect them by a smooth curve. esolutions
2-4 Zeros of Polynomial Functions
List all possible rational zeros of each function Then determine which, if any, are zeros 1 g(x) = x 4 6x 3 31x 2 + 216x 180 Because the leading coefficient is 1, the possible rational zeros are the integer
Practice Test - Chapter 2
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several
2-3 The Remainder and Factor Theorems
Factor each polynomial completely using the given factor and long division. 3. x 3 + 3x 2 18x 40; x 4 So, x 3 + 3x 2 18x 40 = (x 4)(x 2 + 7x + 10). Factoring the quadratic expression yields x 3 + 3x 2
2-5 Rational Functions
19. SALES The business plan for a new car wash projects that profits in thousands of dollars will be modeled by the function p (z) =, where z is the week of operation and z = 0 represents opening. a. State
Practice Test - Chapter 2
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several
2-6 Nonlinear Inequalities
31. Find the domain of each expression. For the expression to be defined, x 2 3x 40 0. Let f (x) = x 2 3x 40. A factored form of f (x) is f (x) = (x 8)(x + 5). f (x) has real zeros at x = 8 and x = 5.
Mid-Chapter Quiz: Lessons 2-1 through 2-3
Graph and analyze each function. Describe its domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 2x 3 Evaluate the function for several
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. 1. 11x 6 5x 5 + 4x 2 degree = 6, leading coefficient = 11 2. 10x 7 5x
10-2 Simplifying Radical Expressions. Simplify each expression. SOLUTION: 4. SOLUTION: SOLUTION: SOLUTION: 10. MULTIPLE CHOICE Which expression is
2. Simplify each expression. 10. MULTIPLE CHOICE Which expression is equivalent to? A B 4. C D 6. 8. 12. The correct choice is D. Simplify each expression. esolutions Manual - Powered by Cognero Page 1
x 2 + 6x 18 x + 2 Name: Class: Date: 1. Find the coordinates of the local extreme of the function y = x 2 4 x.
1. Find the coordinates of the local extreme of the function y = x 2 4 x. 2. How many local maxima and minima does the polynomial y = 8 x 2 + 7 x + 7 have? 3. How many local maxima and minima does the
Chapter 1 Notes: Quadratic Functions
19 Chapter 1 Notes: Quadratic Functions (Textbook Lessons 1.1 1.2) Graphing Quadratic Function A function defined by an equation of the form, The graph is a U-shape called a. Standard Form Vertex Form
Study Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14.
4-6 Study Guide and Intervention Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form ax 2 + bx + c = 0. Quadratic Formula The solutions of
Cumulative Review. Name. 13) 2x = -4 13) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Cumulative Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Evaluate the algebraic expression for the given value or values of the variable(s).
5-5 Solving Polynomial Equations
Factor completely. If the polynomial is not factorable, write prime. 1. 3ax + 2ay az + 3bx + 2by bz (a + b)(3x + 2y z) 2. 2kx + 4mx 2nx 3ky 6my + 3ny (2x 3y)(k + 2m n) 3. 2x 3 + 5y 3 prime 4. 16g 3 + 2h
11.2 The Quadratic Formula
11.2 The Quadratic Formula Solving Quadratic Equations Using the Quadratic Formula. By solving the general quadratic equation ax 2 + bx + c = 0 using the method of completing the square, one can derive
SOLUTION: The domain of a square root function only includes values for which the radicand is nonnegative.
19. Graph each function. State the domain and range. 21. The domain of a square root function only includes values for which the radicand is nonnegative. esolutions Manual - Powered by Cognero Page 1 23.
6-3 Square Root Functions and Inequalities. Identify the domain and range of each function. ANSWER: ANSWER:
Identify the domain and range of each function. 7. 1. 3. Graph each function. State the domain and range. 5. Graph each inequality. 9. esolutions Manual - Powered by Cognero Page 1 11. Graph each function.
b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true
Section 5.2 solutions #1-10: a) Perform the division using synthetic division. b) if the remainder is 0 use the result to completely factor the dividend (this is the numerator or the polynomial to the
Find (f + g )(x), (f g)(x), (x),and (x) for each f (x) and g(x). Indicate any restrictions in
Find (f + g )(x), (f g)(x), (x),and (x) for each f (x) and g(x). Indicate any restrictions in domain or range. 2. esolutions Manual - Powered by Cognero Page 1 For each pair of functions, find and, if
Chapter 3 The Integral Business Calculus 197
Chapter The Integral Business Calculus 97 Chapter Exercises. Let A(x) represent the area bounded by the graph and the horizontal axis and vertical lines at t=0 and t=x for the graph in Fig.. Evaluate A(x)
1) Solve the formula for the indicated variable. P = 2L + 2W for W. 2) Solve the formula for the variable y. 5 = 7x - 8y
Math120 Cumulative Review This is to help prepare you for the 40 question final exam. It is not all inclusive of the material covered in your course. Therefore items not on this review may appear on the
Each element of the domain is paired with exactly one element of the range. So, the relation is a function.
CCSS STRUCTURE State the domain and range of each relation. Then determine whether each relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. 1. The left side
MSLC Math 1075 Final Exam Review. 1. Factor completely Solve the absolute value equation algebraically. g. 8x b. 4x 2 5x. f.
MSLC Math 07 Final Exam Review Disclaimer: This should NOT be used as your only guide for what to study.. Factor completely. a. x y xy xy mn n 7x x x x 0xy x y e. xy y x y f. z z 7 g. mn m n h. c d i.
UNIT 5 QUADRATIC FUNCTIONS Lesson 1: Interpreting Structure in Expressions Instruction
Prerequisite Skills This lesson requires the use of the following skills: evaluating expressions using the order of operations evaluating expressions for a given value identifying parts of an expression
Chapter 2.7 and 7.3. Lecture 5
Chapter 2.7 and 7.3 Chapter 2 Polynomial and Rational Functions 2.1 Complex Numbers 2.2 Quadratic Functions 2.3 Polynomial Functions and Their Graphs 2.4 Dividing Polynomials; Remainder and Factor Theorems
Chapter 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
1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.
Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope
Every polynomial equation of degree 1 or greater has at least one root in the set of complex numbers.
Sec 3.1 Polynomial Functions Fundamental Theorem of Algebra An important and famous German mathematician, Carl Friedrich Gauss, is credited with first proving the FUNDAMENTAL THEOREM OF ALGEBRA which states:
4-3 Solving Quadratic Equations by Factoring. Write a quadratic equation in standard form with the given root(s). 1. 8, 5 ANSWER: ANSWER: ANSWER:
Write a quadratic equation in standard form with the given root(s). 1. 8, 5 8. 9. (2x 5)(x + 6) 2. (4x 3)(4x 1) Solve each equation. 3. 10. 6, 6 4. Factor each polynomial. 5x(7x 3) 11. 12. 5. 6. (6x 1)(3x
8-1 Multiplying and Dividing Rational Expressions. Simplify each expression. ANSWER: 3. MULTIPLE CHOICE Identify all values of x for which
7. 1. 9. 3. MULTIPLE CHOICE Identify all values of x for which is undefined. A 7, 4 B 7, 4 C 4, 7, 7 11. D 4, 7 D 4 5. 13. 15. esolutions Manual - Powered by Cognero Page 1 17. 25. 19. MULTIPLE CHOICE
More 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
Solve each equation by completing the square. Round to the nearest tenth if necessary. 5. x 2 + 4x = 6 ANSWER: 5.2, 1.2
Find the value of c that makes each trinomial a perfect square. 1. x 2 18x + c 81 3. x 2 + 9x + c Solve each equation by completing the square. Round to the nearest tenth if necessary. 5. x 2 + 4x = 6
Just as in the previous lesson, all of these application problems should result in a system of equations with two equations and two variables:
The two methods we have used to solve systems of equations are substitution and elimination. Either method is acceptable for solving the systems of equations that we will be working with in this lesson.
10-1 Sequences as Functions. Determine whether each sequence is arithmetic. Write yes or no. 1. 8, 2, 12, 22
Determine whether each sequence is arithmetic. Write yes or no. 1. 8, 2, 12, 22 Subtract each term from the term directly after it. The common difference is 10. 3. 1, 2, 4, 8, 16 Subtract each term from
S56 (5.1) Polynomials.notebook August 25, 2016
Q1. Simplify Daily Practice 28.6.2016 Q2. Evaluate Today we will be learning about Polynomials. Q3. Write in completed square form x 2 + 4x + 7 Q4. State the equation of the line joining (0, 3) and (4,
MAT 107 College Algebra Fall 2013 Name. Final Exam, Version X
MAT 107 College Algebra Fall 013 Name Final Exam, Version X EKU ID Instructor Part 1: No calculators are allowed on this section. Show all work on your paper. Circle your answer. Each question is worth
Algebra 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
Section 4.2 Polynomial Functions of Higher Degree
Section 4.2 Polynomial Functions of Higher Degree Polynomial Function P(x) P(x) = a degree 0 P(x) = ax +b (degree 1) Graph Horizontal line through (0,a) line with y intercept (0,b) and slope a P(x) = ax
PLC Papers Created For:
PLC Papers Created For: Year 11 Topic Practice Paper: Solving Quadratics (Graphically) Quadratic equations (graphical methods) 1 Grade 6 Objective: Find approximate solutions to quadratic equations using
Need help? Try or 4.1 Practice Problems
Day Date Assignment (Due the next class meeting) Friday 9/29/17 (A) Monday 10/9/17 (B) 4.1 Operations with polynomials Tuesday 10/10/17 (A) Wednesday 10/11/17 (B) 4.2 Factoring and solving completely Thursday
1-4 Extrema and Average Rates of Change
Use the graph of each function to estimate intervals to the nearest 0.5 unit on which the function is increasing, decreasing, or constant. Support the answer numerically. 6. 3. When the graph is viewed
3-1 Constant Rate of Change
Determine whether the relationship between the two quantities shown in the table or graph is linear. If so, find the constant rate of change. If not, explain your reasoning. 1. Analyze the table. The rate
Name: 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
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
8-1 Multiplying and Dividing Rational Expressions. Simplify each expression. SOLUTION: The function is undefined when the denominator tends to 0.
Simplify each expression 3 MULTIPLE CHOICE Identify all values of x for which is undefined 1 A 7, 4 B 7, 4 C 4, 7, 7 D 4, 7 The function is undefined when the denominator tends to 0 The correct choice
Study Guide and Review - Chapter 12
Choose the correct term to complete each sentence. 1. The slope of a nonlinear graph at a specific point is the and can be represented by the slope of the tangent line to the graph at that point. The slope
4x 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,
A repeated root is a root that occurs more than once in a polynomial function.
Unit 2A, Lesson 3.3 Finding Zeros Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial function. This information allows
SECTION 5.1: Polynomials
1 SECTION 5.1: Polynomials Functions Definitions: Function, Independent Variable, Dependent Variable, Domain, and Range A function is a rule that assigns to each input value x exactly output value y =
8-1 Multiplying and Dividing Rational Expressions. Simplify each expression. ANSWER: ANSWER:
Simplify each expression. 1. 2. 3. MULTIPLE CHOICE Identify all values of x for which is undefined. A 7, 4 B 7, 4 C 4, 7, 7 D 4, 7 D Simplify each expression. 4. esolutions Manual - Powered by Cognero
Chapter 8. Exploring Polynomial Functions. Jennifer Huss
Chapter 8 Exploring Polynomial Functions Jennifer Huss 8-1 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial
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 Order of Operations Expression Variable Coefficient
Final Review Accelerated Advanced Algebra
Name: ate: 1. What are the factors of z + z 2 + 25z + 25? 5. Factor completely: (7x + 2) 2 6 (z + 1)(z + 5)(z 5) (z 1)(z + 5i) 2 (49x + 1)(x 8) (7x 4)(7x + 8) (7x + 4)(7x 8) (7x + 4)(x 9) (z 1)(z + 5i)(z
Chapter 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
1-1 Functions < x 64 SOLUTION: 9. { 0.25, 0, 0.25, 0.50, } SOLUTION: 12. all multiples of 8 SOLUTION: SOLUTION:
Write each set of numbers in set-builder and interval notation, if possible. 3. x 4 The set includes all real numbers less than or equal to 4. In set-builder notation this set can be described as {x x
Chapter 2: Polynomial and Rational Functions
Chapter 2: Polynomial and Rational Functions Section 2.1 Quadratic Functions Date: Example 1: Sketching the Graph of a Quadratic Function a) Graph f(x) = 3 1 x 2 and g(x) = x 2 on the same coordinate plane.
Precalculus Chapter 7 Page 1
Section 7.1 Polynomial Functions 1. To evaluate polynomial functions.. To identify general shapes of the graphs of polynomial functions. I. Terminology A. Polynomials in one variable B. Examples: Determine
5-4 Sum and Difference Identities
Find the exact value of each trigonometric expression. 1. cos 75 Write 75 as the sum or difference of angle measures with cosines that you know. 3. sin Write as the sum or difference of angle measures
6.1 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.
10-2 Simplifying Radical Expressions. Simplify each expression. 1. SOLUTION: 2. SOLUTION: 3. SOLUTION: 4. SOLUTION: 5. SOLUTION:
- Simplifying Radical Expressions Simplify each expression.... 4. 5. Page - Simplifying Radical Expressions 5. 6. 7. 8. 9. Page - Simplifying Radical Expressions 9.. MULTIPLE CHOICE Which expression is
5-7 Roots and Zeros. Solve each equation. State the number and type of roots. 1. x 2 3x 10 = 0 ANSWER: 2, 5; 2 real
Solve each equation. State the number and type of roots. 1. x 2 3x 10 = 0 2, 5; 2 real 2. x 3 + 12x 2 + 32x =0 8, 4, 0; 3 real 3. 16x 4 81 = 0 2 real, 2 imaginary 4. 0 = x 3 8 1 real, 2 imaginary State
Functions and Equations
Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario Euclid eworkshop # Functions and Equations c 006 CANADIAN
f(x) = 2x + 5 3x 1. f 1 (x) = x + 5 3x 2. f(x) = 102x x
1. Let f(x) = x 3 + 7x 2 x 2. Use the fact that f( 1) = 0 to factor f completely. (2x-1)(3x+2)(x+1). 2. Find x if log 2 x = 5. x = 1/32 3. Find the vertex of the parabola given by f(x) = 2x 2 + 3x 4. (Give
Algebra II Chapter 5
Algebra II Chapter 5 5.1 Quadratic Functions The graph of a quadratic function is a parabola, as shown at rig. Standard Form: f ( x) = ax2 + bx + c vertex: (x, y) = b 2a, f b 2a a < 0 graph opens down
Mid-Chapter Quiz: Lessons 1-1 through 1-4
Determine whether each relation represents y as a function of x. 1. 3x + 7y = 21 This equation represents y as a function of x, because for every x-value there is exactly one corresponding y-value. function
Factors, Zeros, and Roots
Factors, Zeros, and Roots Solving polynomials that have a degree greater than those solved in previous courses is going to require the use of skills that were developed when we previously solved quadratics.
Study Guide and Review - Chapter 6. Choose a word or term that best completes each statement.
Choose a word or term that best completes each statement. 1. If both compositions result in the,then the functions are inverse functions. identity function 2. In a(n), the results of one function are used
Solving 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
3 What is the degree of the polynomial function that generates the data shown below?
hapter 04 Test Name: ate: 1 For the polynomial function, describe the end behavior of its graph. The leading term is down. The leading term is and down.. Since n is 1 and a is positive, the end behavior
Study Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14.
Study Guide and Intervention Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form a 2 + b + c = 0. Quadratic Formula The solutions of a 2 +
Algebra 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)
3-4 Solving Quadatic Equations by Factoring 15. Divide each side of the equation by 2. Write the equation with the right side equals zero.
15. Divide each side of the equation by 2. Write the equation with the right side equals zero. Use the identity (a b) 2 = a 2 2ab + b 2 to factor the left side of the equation. Here, a = x and b = 6. So,
Unit 2 Rational Functionals Exercises MHF 4UI Page 1
Unit 2 Rational Functionals Exercises MHF 4UI Page Exercises 2.: Division of Polynomials. Divide, assuming the divisor is not equal to zero. a) x 3 + 2x 2 7x + 4 ) x + ) b) 3x 4 4x 2 2x + 3 ) x 4) 7. *)
Chapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions
Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations
Class: Date: Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations 1 Express the following polynomial function in factored form: P( x) = 10x 3 + x 2 52x + 20 2 SE: Express the following
Math Placement Test Review Sheet Louisburg College _ Summer = c = d. 5
1. Preform indicated operations with fractions and decimals: a. 7 14 15 = b. 2 = c. 5 + 1 = d. 5 20 4 5 18 12 18 27 = 2. What is the least common denominator of fractions: 8 21 and 9 14. The fraction 9
Polynomial Functions
Polynomial Functions Equations and Graphs Characteristics The Factor Theorem The Remainder Theorem http://www.purplemath.com/modules/polyends5.htm 1 A cross-section of a honeycomb has a pattern with one
The function is defined for all values of x. Therefore, the domain is set of all real numbers.
Graph each function. State the domain and range. 1. f (x) = 3 x 3 + 2 The function is defined for all values of x. Therefore, the domain is set of all real numbers. The value of f (x) tends to 2 as x tends
2-6 Algebraic Proof. State the property that justifies each statement. 1. If m 1 = m 2 and m 2 = m 3, then m 1 = m 3. SOLUTION:
State the property that justifies each 1. If m 1 = m 2 and m 2 = m 3, then m 1 = m 3. There are two parts to the hypotheses. "If m 1 = m 2 and m 2 = m 3, then m 1 = m 3. "The end of the first part of the
Section 1.3 Rates of Change and Behavior of Graphs
Section 1. Rates of Change and Behavior of Graphs 5 Section 1. Rates of Change and Behavior of Graphs Since functions represent how an output quantity varies with an input quantity, it is natural to ask
Objective Mathematics
Multiple choice questions with ONE correct answer : ( Questions No. 1-5 ) 1. If the equation x n = (x + ) is having exactly three distinct real solutions, then exhaustive set of values of 'n' is given
Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14
Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)
Which 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.
Algebra 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)
2.1 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.
MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions
MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form f x = ax 2 + bx + c,
1. The graph of a quadratic function is shown. Each square is one unit.
1. The graph of a quadratic function is shown. Each square is one unit. a. What is the vertex of the function? b. If the lead coefficient (the value of a) is 1, write the formula for the function in vertex
8-6 Solving Rational Equations and Inequalities. Solve each equation. Check your solution. ANSWER: 11 ANSWER: 9 ANSWER: 7 ANSWER: 3 ANSWER: 8
Solve each equation. Check your solution. 1. 11 2. 9 3. 7 4. 3 5. 8 6. 5 esolutions Manual - Powered by Cognero Page 1 7. 14 8. 14 9. CCSS STRUCTURE Sara has 10 pounds of dried fruit selling for $6.25
Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i
Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat
7-2 Division Properties of Exponents. Simplify each expression. Assume that no denominator equals zero. SOLUTION: SOLUTION: SOLUTION: SOLUTION:
Simplify each expression. Assume that no denominator equals zero. 1. 2. 3. 4. Page 1 4. 5. 6. 7. Page 2 7. 8. 9. 10. Page 3 10. 11. 12. A value to the zero power is 1. 13. A value to the zero power is
2-1 Relations and Functions
CCSS STRUCTURE State the domain and range of each relation. Then determine whether each relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. 4. BASKETBALL
Midterm. Multiple Choice Identify the choice that best completes the statement or answers the question.
Midterm Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Factor completely. If the polynomial cannot be factored, say it is prime. 10x 2-95x + 225 2. Solve
Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).
MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and
Common Core Algebra 2 Review Session 1
Common Core Algebra 2 Review Session 1 NAME Date 1. Which of the following is algebraically equivalent to the sum of 4x 2 8x + 7 and 3x 2 2x 5? (1) 7x 2 10x + 2 (2) 7x 2 6x 12 (3) 7x 4 10x 2 + 2 (4) 12x
Math Practice Exam 2 - solutions
Math 181 - Practice Exam 2 - solutions Problem 1 A population of dinosaurs is modeled by P (t) = 0.3t 2 + 0.1t + 10 for times t in the interval [ 5, 0]. a) Find the rate of change of this population at
Section 2.3 Properties of Functions
22 Section 2.3 Properties of Functions In this section, we will explore different properties of functions that will allow us to obtain the graph of the function more quickly. Objective #1 Determining Even
4-7 Inverse Linear Functions
Find the inverse of each relation. 1. {(4, 15), ( 8, 18), ( 2, 16.5), (3, 15.25)} {( 15, 4), ( 18, 8), ( 16.5, 2), ( 15.25, 3)} 2. {(11.8, 3), (3.7, 0), (1, 1), ( 12.5, 6)} Graph the inverse of each relation.