Chapter 8. Exploring Polynomial Functions. Jennifer Huss


 Barbara Walters
 3 years ago
 Views:
Transcription
1 Chapter 8 Exploring Polynomial Functions Jennifer Huss
2 81 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial functions where the degree is n and a is the coefficient look like this: f(x) = a 0 x n + a 1 x n1 + +a n1 x + a n Example: f(x) = 4x 4 3x x 2 7x + 2 (degree 4) If we know that x = 5 then f(5) replaces each x in the polynomial with a 5 The degree of the function tells the maximum number of real zeros the function has, or the number of times the graph of the function crosses the xaxis (ex: degree 4 function means there are at most 4 real zeros) A leading coefficient is the coefficient on the term of highest degree, in the example above it would be 4 because 4x 4 is the term of highest degree See the book for more about the functions of different degrees
3 81 Polynomial Functions (cont.) Even functions (degrees 0, 2, 4, 6, etc.) take this form: Both sides of this graph go up (+) or both go down () Odd functions (degrees 1, 3, 5, 7, etc.) take this form: One side of this graph rises (+) and the other side falls ()
4 81 Examples 1. Determine if this expression is a polynomial in one variable. If it is, give the degree of the function. a) X 2 + 2xy + y 2 This is not a function because it has x and y variables. b) 2a 2 2a + 4 This is a polynomial and it s degree is 2. c) 12 2/n + n 2 This is not a polynomial because 2/n has a negative degree. d) c c 6 This is a polynomial of degree Find p(m + 2) if p(x) = 3x 8x 2 + x 3. p(m + 2) = 3(m + 2) 8(m + 2) 2 + (m + 2) 3 = 3m + 6 8(m 2 + 4m + 4) + (m + 2)(m 2 + 4m + 4) = 3m + 6 8m 2 32m 32 + m 3 + 6m m + 8 = m 3 2m 2 17m 18
5 81 Examples (cont.) 3. Decide if the graph is even or odd and tell how many real zeros it has. 1. f(x) = x 3 5x f(x) = x 4 3x This is an odd function with 3 real zeros. This is an even function with 2 real zeros.
6 1) 505 2) x 2 + 2x + 5 3) even function, 4 real zeros 81 Problems 1. Find f(3) for f(x) = x 5 + 5x 4 15x Find f(x + 2) for f(x) = x 2 2x Graph f(x) = x 4 5x Decide if its an even or odd function and tell how many real zeros it has.
7 82 The Remainder and Factor Theorems There are two parts of the remainder theorem: 1. If the polynomial f(x) is divided by (x a), the remainder will be a number that is equal to f(a) 1. I.e.. If f(x) is divided by x 4, f(4) will give the value of the remainder 2. Dividend = (quotient x divisor) + remainder 1. also can see this as f(x) = [q(x) x (x a)] + f(a) 2. The quotient is always a polynomial with one degree less than f(x) Synthetic division is helpful in solving these problems (this can also be called synthetic substitution) Factor theorem: (x a) is a factor of f(x) if and only if the remainder (or f(a)) is equal to zero This is a good way to find the first factor of a polynomial The quotient may also be called a depressed polynomial because it has one less degree than the original polynomial
8 82 Examples 1. Use synthetic division and direct substitution to find f(4) when f(x) = x 4 6x 3 + 8x 2 + 5x f(4) = 4 4 6(4) 3 + 8(4) 2 + 5(4) OR = f(4) = Give the factors of x 3 11x x 36 if one factor is x So, after we divide the polynomial by x we are left with x 2 5x + 6 which we can solve by factoring into (x 3)(x + 2). This means the factors are (x 6), (x 3), and (x + 2). This can also be written in the f(x) = quotient x divisor + remainder. This would look like f(x) = (x 2 5x + 6)(x 6) + 0.
9 1) (4x 2 + 3x 1)(x 3) 5 2) f(5) = 63 3) (x + 5), (x 2), and (x + 3) 82 Problems 1. Use synthetic division to do (4x 3 9x 2 10x 2) divided by (x 3). Then write the answer in the form f(x) = [quotient x divisor] + remainder. 2. Given f(x) = 4x 2 + 6x 7, find f(5) by synthetic division or direct substitution. 3. Five the factors of x 3 + 6x 2 x 30 if one factors is (x + 5).
10 83 Graphing Polynomial Functions and Approximating Zeros Look back at 81 to help with understanding finding zeros and the definition of even and odd functions Location Principle: If y = f(x) is a polynomial function and you have a and b such that f(a) < 0 and f(b) > 0 then there will be some number in between a and b that is a zero of the function b a zero A relative maximum is the highest point between two zeros and a relative minimum is the lowest point between two zeros
11 83 Example Graph the function f(x) = 2x 3 5x 2 + 3x + 2 and approximate the real zeros. There are zeros at approximately 2.9, 0.4, and 0.8.
12 1) The real zeros are approximately 2, 1, and Problem 1. Graph f(x) = x 3 + x 2 4x 4 and approximate the real zeros. Show the relative minimum and maximum on the graph.
13 84 Roots and Zeros The Fundamental Theorem of Algebra says that every polynomial equation has at least one root in the set of complex numbers Another way to state it: a polynomial with degree n has exactly n roots in the set of complex numbers Remember: roots can be imaginary (complex numbers) The Complex Conjugates Theorem says that if a + bi is a zero of a polynomial function then a bi is also a zero of the function Descartes Rule of Signs says that if f(x) is a polynomial with its terms arranged in order of decreasing power (ex: x 3, x 2, x) then: The number of positive real zeros is given by the number of sign changes of the coefficients of f(x), or less than the number of sign changes by an even number The number of negative real zeros is given by the number of sign changes of the coefficients of f(x), or less than the number of sign changes by an even number Ex: 5 sign changes for f(x) means 5, 3, or 1 positive real zeros
14 84 Examples 1. Give the possible number of positive real zeros, negative real zeros, and imaginary zeros of f(x) = x 3 7x x 10. Then find all the zeros if one zero is 3 i. f(x) = x 3 7x x 10 3 sign changes, so 3 or 1 positive real zeros f(x) = x 3 7x 2 16x 10 0 sign changes, so no negative real zeros Since the degree is 3 on this polynomial we should have 3 zeros. If we have 3 positive real zeros there will be no imaginary zeros. If we have 1 positive real zero there will be 2 imaginary zeros. So, 3 positive real zeros or 1 positive real zero and 2 imaginary zeros. Since 3 i is one zero, 3 + i will also be a zero. f(x) = [x (3 i)][x (3 + i)](?) f(x) = [ x 2 (3 i)x (3 + i)x + (3 i)(3 + i)](?) f(x) = (x 2 3x + xi 3x xi + 9 i 2 ) (?) f(x) = (x 2 6x + 10) (?) So now we need to find the (?), which is the third factor, by long division. x 1 x 2 6x + 10 ) x 3 7x x 10 (x 3 6x x) x 2 + 6x 10 (x 2 + 6x 10) 0 _ So, (x 1) is the third factor, which means the third zero is 1. The zeros are 3 + i, 3 i, and 1.
15 84 Examples (cont.) 2. Given that 1 and 1 + i are two zeros of a polynomial, write the polynomial of the least degree having these zeros. If 1 + i is a zero, 1 i is another zero. f(x) = [x (1 + i)] [x (1 i)] (x 1) f(x) = [ x 2 (1 + i)x (1 i)x + (1 + i)(1 i)] (x 1) f(x) = [x 2 x xi x + xi + 1 i 2 ] (x 1) f(x) = (x 2 2x + 2) (x 1) f(x) = x 3 2 x 2 + 2x x 2 + 2x 2 f(x) = x 3 3x 2 + 4x 2 The polynomial is x 3 3x 2 + 4x 2.
16 1) 2 or 0 positive real zeros, 1 negative real zero, 2 or 0 imaginary zeros. 2) The zeros are 1 3i, 1 + 3i, and Problems 1. State the number of positive real zeros, negative real zeros, and imaginary zeros in f(x) = 16x 3 + 6x 2 7x Given f(x) = x 3 + 6x + 20 and one of its zeros as 1 3i, find all of the zeros of this function.
17 85 Rational Zero Theorem The rational zero theorem helps us find zeros when we have large numbers that are hard to factor Rational Zero Theorem says that if you have a polynomial f(x) = a 0 x n + + a n1 x + a n, then you can find zeros by doing p divided by q if p is a factor of a n and q is a factor of a o A similar theorem, the Integral Zero Theorem, says that if a 0 = 1 and a n = 0, then q = 1 which makes p/q= p. This means that all the zeros of this function will simply be the factors of a n. To find which zeros actually work, you need to do the Descartes Rule of Signs and graph the function
18 85 Example List the possible rational zeros for f(x) = 3x 4 2x 3 5. Then graph the function to see which are the actual rational zeros. a 0 = 3 which means q = 1, 3 a n = 5 which means p = 1, 5 Possible rational zeros are: 1, 5, 1, 5 or 1, 5, 1/3, 5/ The real zeros are 1 and 5/3.
19 1) 1, 2, 7, 4 2) The zeros are 2 and Problems 1. List the possible rational zeros of f(x) = x 4 8x 3 + 7x Find the rational zeros of f(x) = x 3 x 2 8x + 12.
20 86 Using Quadratic Techniques to Solve Polynomial Equations Sometimes we want to solve or factor a polynomial that is not degree 2 (x 2 ) We try to force the polynomial into the quadratic form so then we can factor and solve it The quadratic form is: a[f(x)] 2 + b[f(x)] + c = 0 This is a variation of ax 2 + bx + c = 0 where our x term could change depending on the problem
21 86 Examples Solve the following equations. 1) x 4 7x = 0 2) t = 0 (x 2 ) 2 7(x 2 ) + 12 = 0 First, you must look at the graph to find the (x 2 4)(x 2 3)= 0 first zero at x = 6. Then perform long division. x 2 4 = 0 x 2 3 = 0 x 2 = 4 x 2 = 3 t 2 + 6t + 36 x = 4 x = 3 t 6 t x = 2 The solutions or zeros are This gives (t 6)(t 2 + 6t + 36) = 0. 2, 2, 3, and  3. t 2 + 6t + 36 can t be factored so we use the quadratic formula. t =  6 (6) 2 4(1)(36) = 3 3i 3 2(1) The zeros are 6, i 3, and 3 3i 3.
22 86 Examples (cont.) 3. y 8 y + 7 = 0 ( y) 2 8( y) + 7 = 0 ( y 7)( y 1) = 0 y 7 = 0 y 1 = 0 y = 7 y = 1 y = 49 y = 1 The solutions or zeros are 1 and 49.
23 1) 16 and 81 2) 2, 2, 2,and  2 3) 0, 4, and Problems Solve each equation. 1. s 13 s + 36 = 0 2. x 4 6x 2 = n n n = 0
24 87 Composition of Functions The composition of functions is when you combine two functions to create one multistep function The composition function f g needs to have the range of g as part of the domain of f (the output of g is part of the input for f) The composition f g is written as f[g(x)] In these problems you solve g(x) to get some value a, and then you solve f(a) to get the final answer Two functions may not have a composition if we find g(x) to be a, but f(a) is not possible Iteration is a special composition where the function combines with itself, for example, f[f(x)]
25 87 Examples 1) If f = {(1, 4) (10, 5) (6, 3)} and g = {(5, 1) (4, 6)} then find f g. f g 5 4 Domain (x s) of g f[g(5)] = f(1) = 4 f[g(4)] = f(6) = Range (y s) of g Domain (x s) of f Range (y s) of f 2) If f(x) = x + 7 and g(x) = x 2 4, find [f g](2) and [g f](2). [f g](2) = f[g(2)] [g f](2) = g[f(2)] = f(2 2 4) = g(2 + 7) = f(44) = g(9) = f(0) = (9) 2 4 = = 81 4 = 7 = 77
26 1) [f g] is 16 and [g f] is 195 2) f[g(x)] = 8 6x 87 Problems 1. If f(x) = 2x + 10 and g(x) = x 2 1, find [f g](2) and [g f](2). 2. If f(x) = 8 2x and g(x) = 3x, find f[g(x)].
27 88 Inverse Functions and Relations Two functions, f and g, are inverse functions (opposites) if their composition gives the identity function (x) [f g](x) = x and [g f](x) = x To check for inverses, take both compositions and see if both equal x Also, if you graph the functions the inverse functions should be mirror images or reflections of one another across the line y = x f 1 mean f inverse and f = g 1 means f is the inverse of g If f and f 1 are inverse functions, f(a) = b and f 1 (b) = a This means that the ordered pair (a, b) will change to (b, a) for the inverse function To write an inverse function, switch the x and the y of the equation y = ax + b changes to x = ay + b Inverse relations means that a relation (set of ordered pairs) can be changed into an inverse by switching (a, b) to (b, a)
28 88 Examples 1. Determine whether f(x) = 6 2x and g(x) = ½(6 x) are inverse functions. Check by graphing. In order to determine this we will find [f g](x) and [g f](x). g(x) f(x) [f g](x) = f[g(x)] [g f](x) = g[f(x)] = f [1/2(6 x)] = g(6 2x) = 6 2[1/2(6 x)] = ½[6 (6 2x)] = x = ½ ( x) = x = ½ (2x) Yes, f(x) and g(x) = x are mirror images. Yes, they are inverse functions since both compositions equal x and the graphs are mirror images.
29 88 Examples (cont.) 2. Find the inverse of f(x) = x + 3. Then graph both functions to verify they are inverses. To find the inverse, switch y and x. f(x) = x + 3 y = x + 3 x = y + 3 y = x 3 f 1 = x 3 The graphs are mirror images across y = x. Check: [f f 1 ](x) = f(x 3) [f 1 f](x) = f 1 (x + 3) = (x 3) + 3 = (x + 3) 3 = x = x Yes, f 1 = x 3 is the inverse function.
30 1) f 1 = (1/2)x (5/2) 2) No 88 Problems 1. Find the inverse of f(x) = 2x + 5 and graph the function and the inverse function. 2. Determine if f(x) = 3x 9 and g(x) = 3x + 9 are inverse functions.
31 88B Square Root Functions and Relations Square root functions can never be negative if we want to find answers that are real numbers The square root graph looks like the following: y = x For examples and practice problems, see the textbook
Math 110 Midterm 1 Study Guide October 14, 2013
Name: For more practice exercises, do the study set problems in sections: 3.4 3.7, 4.1, and 4.2. 1. Find the domain of f, and express the solution in interval notation. (a) f(x) = x 6 D = (, ) or D = R
More information6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4
2.3 Real Zeros of Polynomial Functions Name: Precalculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using
More informationPolynomial Functions
Polynomial Functions Polynomials A Polynomial in one variable, x, is an expression of the form a n x 0 a 1 x n 1... a n 2 x 2 a n 1 x a n The coefficients represent complex numbers (real or imaginary),
More informationRoots & Zeros of Polynomials. How the roots, solutions, zeros, xintercepts and factors of a polynomial function are related.
Roots & Zeros of Polynomials How the roots, solutions, zeros, xintercepts and factors of a polynomial function are related. A number a is a zero or root of a function y = f (x) if and only if f (a) =
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 informationMAT116 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 seconddegree polynomial. Standard Form f x = ax 2 + bx + c,
More informationDividing Polynomials: Remainder and Factor Theorems
Dividing Polynomials: Remainder and Factor Theorems When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is zero, then the divisor is a factor of the dividend.
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 informationModeling Data. 27 will get new packet. 24 Mixed Practice 3 Binomial Theorem. 23 Fundamental Theorem March 2
Name: Period: PreCal AB: Unit 1: Polynomials Monday Tuesday Block Friday 11/1 1 Unit 1 TEST Function Operations and Finding Inverses 16 17 18/19 0 NO SCHOOL Polynomial Division Roots, Factors, Zeros and
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 informationOperations w/polynomials 4.0 Class:
Exponential LAWS Review NO CALCULATORS Name: Operations w/polynomials 4.0 Class: Topic: Operations with Polynomials Date: Main Ideas: Assignment: Given: f(x) = x 2 6x 9 a) Find the yintercept, the equation
More informationZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS
ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS TOOLS IN FINDING ZEROS OF POLYNOMIAL FUNCTIONS Synthetic Division and Remainder Theorem (Compressed Synthetic Division) Fundamental
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 informationWarmUp. Simplify the following terms:
WarmUp Simplify the following terms: 81 40 20 i 3 i 16 i 82 TEST Our Ch. 9 Test will be on 5/29/14 Complex Number Operations Learning Targets Adding Complex Numbers Multiplying Complex Numbers Rules for
More informationA Partial List of Topics: Math Spring 2009
A Partial List of Topics: Math 112  Spring 2009 This is a partial compilation of a majority of the topics covered this semester and may not include everything which might appear on the exam. The purpose
More informationReview 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
More informationPolynomial and Rational Functions. Chapter 3
Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) 30
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 informationPreCalculus: Semester 1 Final Exam Review
Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain
More informationChapter 2. Polynomial and Rational Functions. 2.5 Zeros of Polynomial Functions
Chapter 2 Polynomial and Rational Functions 2.5 Zeros of Polynomial Functions 1 / 33 23 Chapter 2 Homework 2.5 p335 6, 8, 10, 12, 16, 20, 24, 28, 32, 34, 38, 42, 46, 50, 52 2 / 33 23 3 / 33 23 Objectives:
More informationLesson 2.1: Quadratic Functions
Quadratic Functions: Lesson 2.1: Quadratic Functions Standard form (vertex form) of a quadratic function: Vertex: (h, k) Algebraically: *Use completing the square to convert a quadratic equation into standard
More informationTable of contents. Polynomials Quadratic Functions Polynomials Graphs of Polynomials Polynomial Division Finding Roots of Polynomials
Table of contents Quadratic Functions Graphs of Polynomial Division Finding Roots of Jakayla Robbins & Beth Kelly (UK) Precalculus Notes Fall 2010 1 / 65 Concepts Quadratic Functions The Definition of
More informationChapter 3: Polynomial and Rational Functions
Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers
More informationStudent: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed
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 n1,, a 2, a 1, a 0 be real
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 informationUMUC MATH107 Final Exam Information
UMUC MATH07 Final Exam Information What should you know for the final exam? Here are some highlights of textbook material you should study in preparation for the final exam. Review this material from
More informationDownloaded from
Question 1: Exercise 2.1 The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) Page 1 of 24 (iv) (v) (v) Page
More informationx 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
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 informationPreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College
PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions
More information3.4. ZEROS OF POLYNOMIAL FUNCTIONS
3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find
More informationChapter 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
More information, a 1. , a 2. ,..., a n
CHAPTER Points to Remember :. Let x be a variable, n be a positive integer and a 0, a, a,..., a n be constants. Then n f ( x) a x a x... a x a, is called a polynomial in variable x. n n n 0 POLNOMIALS.
More informationChapter 2 Formulas and Definitions:
Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)
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 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 informationSkills Practice Skills Practice for Lesson 10.1
Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).
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 informationPreAlgebra 2. Unit 9. Polynomials Name Period
PreAlgebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (noncomplex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:
More informationOctober 28, S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationSynthetic Division. Vicky Chen, Manjot Rai, Patricia Seun, Sherri Zhen S.Z.
Synthetic Division By: Vicky Chen, Manjot Rai, Patricia Seun, Sherri Zhen S.Z. What is Synthetic Division? Synthetic Division is a simpler way to divide a polynomial by a linear factor. You can consider
More informationA 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 xintercepts of a polynomial function. This information allows
More informationAlgebra 2 Segment 1 Lesson Summary Notes
Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the
More informationChapter 7 Algebra 2 Honors 1 Polynomials
Chapter 7 Algebra 2 Honors 1 Polynomials Polynomial:   Polynomials in one variable Degree Leading coefficient f(x) = 3x 3 2x + 4 f(2) = f(t) = f(y 1) = 3f(x) = Using your graphing calculator sketch/graph
More information171S4.4 Theorems about Zeros of Polynomial Functions. March 27, 2012
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationSection 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
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational
More informationChapter 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.
More informationMaths Class 11 Chapter 5 Part 1 Quadratic equations
1 P a g e Maths Class 11 Chapter 5 Part 1 Quadratic equations 1. Real Polynomial: Let a 0, a 1, a 2,, a n be real numbers and x is a real variable. Then, f(x) = a 0 + a 1 x + a 2 x 2 + + a n x n is called
More information24 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
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 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
51 Polynomial Functions Objectives A2.A.APR.A.2 (formerly AAPR.A.3) Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function
More informationWe say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:
R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)
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 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 informationAlgebra 2 Honors: Final Exam Review
Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt
More informationProcedure for Graphing Polynomial Functions
Procedure for Graphing Polynomial Functions P(x) = a nx n + a n1x n1 + + a 1x + 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
More informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 21 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 informationChapter 31 Polynomials
Chapter 3 notes: Chapter 31 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 informationSolving Quadratic Equations by Formula
Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always
More informationSection 4.3. Polynomial Division; The Remainder Theorem and the Factor Theorem
Section 4.3 Polynomial Division; The Remainder Theorem and the Factor Theorem Polynomial Long Division Let s compute 823 5 : Example of Long Division of Numbers Example of Long Division of Numbers Let
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 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 informationA quadratic expression is a mathematical expression that can be written in the form 2
118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is
More informationLearning Objectives. Zeroes. The Real Zeros of a Polynomial Function
The Real Zeros of a Polynomial Function 1 Learning Objectives 1. Use the Remainder and Factor Theorems 2. Use the Rational Zeros Theorem to list the potential rational zeros of a polynomial function 3.
More informationChapter REVIEW ANSWER KEY
TEXTBOOK HELP Pg. 313 Chapter 3.23.4 REVIEW ANSWER KEY 1. What qualifies a function as a polynomial? Powers = nonnegative integers Polynomial functions of degree 2 or higher have graphs that are smooth
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 3: Polynomial and Rational Functions
Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions A polynomial on degree n is a function of the form P(x) = a n x n + a n 1 x n 1 + + a 1 x 1 + a 0, where n is a nonnegative integer
More informationSection 1.3 Review of Complex Numbers
1 Section 1. Review of Complex Numbers Objective 1: Imaginary and Complex Numbers In Science and Engineering, such quantities like the 5 occur all the time. So, we need to develop a number system that
More informationClass IX Chapter 2 Polynomials Maths
NCRTSOLUTIONS.BLOGSPOT.COM Class IX Chapter 2 Polynomials Maths Exercise 2.1 Question 1: Which of the following expressions are polynomials in one variable and which are No. It can be observed that the
More informationPolynomial Functions and Models
1 CAFall 2011Jordan 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 informationCh. 12 Higher Degree Equations Rational Root
Ch. 12 Higher Degree Equations Rational Root Sec 1. Synthetic Substitution ~ Division of Polynomials This first section was covered in the chapter on polynomial operations. I m reprinting it here because
More informationQuestion 1: The graphs of y = p(x) are given in following figure, for some Polynomials p(x). Find the number of zeroes of p(x), in each case.
Class X  NCERT Maths EXERCISE NO:.1 Question 1: The graphs of y = p(x) are given in following figure, for some Polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) (iv) (v)
More informationLesson 19 Factoring Polynomials
Fast Five Lesson 19 Factoring Polynomials Factor the number 38,754 (NO CALCULATOR) Divide 72,765 by 38 (NO CALCULATOR) Math 2 Honors  Santowski How would you know if 145 was a factor of 14,436,705? What
More informationNAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 11) 3a 2 b 4, given a = 3, b = 2
51 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 information1. Definition of a Polynomial
1. Definition of a Polynomial What is a polynomial? A polynomial P(x) is an algebraic expression of the form Degree P(x) = a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 3 x 3 + a 2 x 2 + a 1 x + a 0 Leading
More information171S4.3 Polynomial Division; The Remainder and Factor Theorems. October 26, Polynomial Division; The Remainder and Factor Theorems
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationCONTENTS COLLEGE ALGEBRA: DR.YOU
1 CONTENTS CONTENTS Textbook UNIT 1 LECTURE 11 REVIEW A. p. LECTURE 1 RADICALS A.10 p.9 LECTURE 1 COMPLEX NUMBERS A.7 p.17 LECTURE 14 BASIC FACTORS A. p.4 LECTURE 15. SOLVING THE EQUATIONS A.6 p.
More information171S4.3 Polynomial Division; The Remainder and Factor Theorems. March 24, Polynomial Division; The Remainder and Factor Theorems
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information2.5 Complex Zeros and the Fundamental Theorem of Algebra
210 CHAPTER 2 Polynomial, Power, and Rational Functions What you ll learn about Two Major Theorems Complex Conjugate Zeros Factoring with Real Number Coefficients... and why These topics provide the complete
More informationPolynomial Expressions and Functions
Hartfield College Algebra (Version 2017a  Thomas Hartfield) Unit FOUR Page  1  of 36 Topic 32: Polynomial Expressions and Functions Recall the definitions of polynomials and terms. Definition: A polynomial
More information1) Synthetic Division: The Process. (Ruffini's rule) 2) Remainder Theorem 3) Factor Theorem
J.F. Antona 1 Maths Dep. I.E.S. Jovellanos 1) Synthetic Division: The Process (Ruffini's rule) 2) Remainder Theorem 3) Factor Theorem 1) Synthetic division. Ruffini s rule Synthetic division (Ruffini s
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 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 informationSect Complex Numbers
161 Sect 10.8  Complex Numbers Concept #1 Imaginary Numbers In the beginning of this chapter, we saw that the was undefined in the real numbers since there is no real number whose square is equal to a
More information5.1 Monomials. Algebra 2
. Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific
More informationAnalysis of Polynomial & Rational Functions ( summary )
Analysis of Polynoial & Rational Functions ( suary ) The standard for of a polynoial function is ( ) where each of the nubers are called the coefficients. The polynoial of is said to have degree n, where
More informationDay 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 22x x 29x x 2 + 6x + 5
Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 22x  15 2. x 29x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There
More informationMath 0320 Final Exam Review
Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:
More informationMath 0312 EXAM 2 Review Questions
Name Decide whether the ordered pair is a solution of the given system. 1. 4x + y = 2 2x + 4y = 20 ; (2, 6) Solve the system by graphing. 2. x  y = 6 x + y = 16 Solve the system by substitution. If
More informationComplex 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
More informationStudy Guide for Math 095
Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.
More informationSection 3.1 Quadratic Functions
Chapter 3 Lecture Notes Page 1 of 72 Section 3.1 Quadratic Functions Objectives: Compare two different forms of writing a quadratic function Find the equation of a quadratic function (given points) Application
More informationSECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
More informationUnit 4 Polynomial/Rational Functions Zeros of Polynomial Functions (Unit 4.3)
Unit 4 Polynomial/Rational Functions Zeros of Polynomial Functions (Unit 4.3) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Find
More information3 Inequalities Absolute Values Inequalities and Intervals... 18
Contents 1 Real Numbers, Exponents, and Radicals 1.1 Rationalizing the Denominator................................... 1. Factoring Polynomials........................................ 1. Algebraic and Fractional
More informationUnit 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. *)
More informationSB CH 2 answers.notebook. November 05, Warm Up. Oct 8 10:36 AM. Oct 5 2:22 PM. Oct 8 9:22 AM. Oct 8 9:19 AM. Oct 8 9:26 AM.
Warm Up Oct 8 10:36 AM Oct 5 2:22 PM Linear Function Qualities Oct 8 9:22 AM Oct 8 9:19 AM Quadratic Function Qualities Oct 8 9:26 AM Oct 8 9:25 AM 1 Oct 8 9:28 AM Oct 8 9:25 AM Given vertex (1,4) and
More informationb) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true
Section 5.2 solutions #110: 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
More information