(2) Dividing both sides of the equation in (1) by the divisor, 3, gives: =

Size: px
Start display at page:

Download "(2) Dividing both sides of the equation in (1) by the divisor, 3, gives: ="

Transcription

1 Dividing Polynomials Prepared by: Sa diyya Hendrickson Name: Date: Let s begin by recalling the process of long division for numbers. Consider the following fraction: Recall that fractions are just division problems, so we proceed with the process of long division small We can express our result in one of the following ways: (1) 2683 = 3(894) + 1 (2) Dividing both sides of the equation in (1) by the divisor, 3, gives: = Now, let s consider a fraction, where the numerator and denominator are polynomials. P (x) D(x) = 2x3 + 5x 2 6x + 7 x 2 2x 5 The good news is that the strategies for dividing polynomials are almost identical to those used for numbers! Before exploring an example, we first make note of the Division Algorithm for polynomials. Making Math Possible 1 of 6 c Sa diyya Hendrickson

2 Dividing Polynomials Division Algorithm: If P (x) and D(x) are polynomials, with D(x) = 0, then there exist unique polynomials Q(x) and R(x), where R(x) is either 0 or of degree less than the degree of D(x), such that P (x) R(x) = Q(x) + D(x) D(x) or equivalently, P (x) = D(x) Q(x) + R(x) When the remainder equals zero, we say that D(x) is a factor of the polynomial, P (x). Notice that this is identical to the definition of a factor in the context of integers! Definition: zero/root of a polynomial c R is a zero/root of a polynomial P (x) if P (c) = 0. (e.g. x = 2 is a root of p(x) = x 2 5x + 6.) Exercise 1: Verify that t = is a root of p(t) = t 2 + 2t 1 Sample solution: By definition of a root, we must verify that p( 1 + 2) = 0. p( 1 + 2) = ( 1 + 2) 2 + 2( 1 + 2) 1 = ( 1 + 2)( 1 + 2) = ( 2) = = 0, as required. It is important to note that some polynomials do not have any zeros. For example, because x 2 0 for all x R, p(x) = x always produces values that are greater than or equal to. However, if a polynomial has least one root, there is an interesting and useful relationship between the root and a factor of the polynomial. This relationship is formally stated in the Factor Theorem, given on the next page. Making Math Possible 2 of 6 c Sa diyya Hendrickson

3 Factor Theorem Factor Theorem: Let P (x) denote a polynomial. Then, c is a zero of P (x) if and only if x c is a factor of P (x). The Factor Theorem gives us the following results: 1. If x = c is a zero of P (x), then x c is a factor of P (x). 2. If x c is a factor of P (x), then x = c is a zero of P (x). Exercise 2: Consider the polynomial p(x) = 2x 2 + x 6. Verify that x = 3 2 are roots. What can we then conclude and why? and x = 2 Sample solution: We must verify that p 3 2 = 0 = p( 2) p = = = = = 6 6 = and p ( 2) = 2 ( 2) 2 + ( 2) 6 = 2 (4) 2 6 = 8 8 = 0 Because x = 3 2 and x = 2 are roots, we conclude (by the Factor Theorem) that x 3 2 and x ( 2) = x + 2 are factors. Notice that if we factor by trial and error, we have: 2x 2 + x 6 = (2x 3)(x + 2) 2 = 2x 3 2 = 2 x 3 2 By factoring out 2, we can see the factors! (x + 2) (x + 2) Making Math Possible 3 of 6 c Sa diyya Hendrickson

4 Polynomial Long Division Exercise 3: Suppose we are given the following information about a polynomial: Degree = 2 Roots: x = 1 ± 2 Leading Coefficient = 3 Determine the expanded and factored forms of the polynomial. Sample solution: Because the polynomial is of degree 2, it can be expressed as: p(x) = ax 2 + bx + c, where a, b, c R Furthermore, since x = 1 ± 2 are roots, the Factor Theorem gives that x ( 1 + 2) and x ( 1 2) are factors. Thus, the factored form must be: p(x) = a (x ( 1 + 2))(x ( 1 2)) To achieve a leading coefficient of 3, we require that a = 3 (Why?). For expanded form: p(x) = 3(x ( 1 + 2))(x ( 1 2)) = 3(x 2 ( 1 2)x ( 1 + 2)x + ( 1 + 2)( 1 2)) = 3(x 2 ( )x + ( 1) 2 ( 2) 2 ) = 3(x 2 ( 2)x + 1 2) = 3(x 2 + 2x 1) = 3x 2 + 6x 3 There are two popular methods for dividing polynomials: long division and synthetic division. Synthetic division is an approach that tends to be easier on the eyes and more compact, but it has its shortcomings. In particular, synthetic division can only be used when D(x) is linear. On the other hand, polynomial long division has no such limitations! So, in this handout, we will explore the more robust method of polynomial long division. Let s begin with the following exercise: Exercise 4: Find the quotient and remainder of x 4 16 x + 2 Option 1: Notice that x 4 16 = x (2 2) 4 2 P.O.E = (x 2 ) = (x 2 + 4)(x 2 4) factoring D.O.S. = (x 2 + 4)(x ) = (x 2 + 4)(x + 2)(x 2) factoring D.O.S. Now, we can see that x + 2 is a factor of x Thus, the remainder is zero. Furthermore, the quotient is the product of the remaining factors: (x 2 + 4)(x 2) = x 3 2x 2 + 4x 8. Making Math Possible 4 of 6 c Sa diyya Hendrickson

5 Polynomial Long Division Exercise 4 (Continued) Option 2: To find the quotient and remainder, we consider polynomial long division: The quotient is the cubic polynomial, x 3 2x 2 + 4x 8 and the remainder is zero. Option 3: Notice that x = ±2 are roots of p(x) = x 4 16, since: p(±2) = (±2) 2 16 = = 0 Thus, by the Factor Theorem, x 2 and x ( 2) = x+2 are factors. Furthermore, because x + 2 is a factor, we know that the remainder is zero. Moreover, (x 2)(x + 2) = x 2 4 must be a quadratic factor of p(x). To determine the remaining factor, consider long division: The the remainder is zero and the quotient is the product of the factors: x 2 and x This product gives the cubic polynomial, x 3 2x 2 + 4x 8. Making Math Possible 5 of 6 c Sa diyya Hendrickson

6 Polynomial Long Division Exercise 5: Express 6x4 2x 3 + x 5 3x 3 + x 1 as Q(x) + R(x) D(x). Sample solution: By long division, we have: Making Math Possible 6 of 6 c Sa diyya Hendrickson

Downloaded from

Downloaded 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 information

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.

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. 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 information

Read the following definitions and match them with the appropriate example(s) using the lines provided.

Read the following definitions and match them with the appropriate example(s) using the lines provided. Algebraic Expressions Prepared by: Sa diyya Hendrickson Name: Date: Read the following definitions and match them with the appropriate example(s) using the lines provided. 1. Variable: A letter that is

More information

Class IX Chapter 2 Polynomials Maths

Class 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 information

Warm-Up. Use long division to divide 5 into

Warm-Up. Use long division to divide 5 into Warm-Up Use long division to divide 5 into 3462. 692 5 3462-30 46-45 12-10 2 Warm-Up Use long division to divide 5 into 3462. Divisor 692 5 3462-30 46-45 12-10 2 Quotient Dividend Remainder Warm-Up Use

More information

Math 1310 Section 4.1: Polynomial Functions and Their Graphs. A polynomial function is a function of the form ...

Math 1310 Section 4.1: Polynomial Functions and Their Graphs. A polynomial function is a function of the form ... Math 1310 Section 4.1: Polynomial Functions and Their Graphs A polynomial function is a function of the form... where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function

More information

How might we evaluate this? Suppose that, by some good luck, we knew that. x 2 5. x 2 dx 5

How might we evaluate this? Suppose that, by some good luck, we knew that. x 2 5. x 2 dx 5 8.4 1 8.4 Partial Fractions Consider the following integral. 13 2x (1) x 2 x 2 dx How might we evaluate this? Suppose that, by some good luck, we knew that 13 2x (2) x 2 x 2 = 3 x 2 5 x + 1 We could then

More information

PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces.

PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces. PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION NOAH WHITE The basic aim of this note is to describe how to break rational functions into pieces. For example 2x + 3 1 = 1 + 1 x 1 3 x + 1. The point is that

More information

( 3) ( ) ( ) ( ) ( ) ( )

( 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 information

6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4

6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4 2.3 Real Zeros of Polynomial Functions Name: Pre-calculus. 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 information

4 Unit Math Homework for Year 12

4 Unit Math Homework for Year 12 Yimin Math Centre 4 Unit Math Homework for Year 12 Student Name: Grade: Date: Score: Table of contents 3 Topic 3 Polynomials Part 2 1 3.2 Factorisation of polynomials and fundamental theorem of algebra...........

More information

CHAPTER 2 POLYNOMIALS KEY POINTS

CHAPTER 2 POLYNOMIALS KEY POINTS CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x

More information

Chapter 2 notes from powerpoints

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

More information

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 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 information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 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 information

Chapter 2 Formulas and Definitions:

Chapter 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 information

PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces.

PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces. PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION NOAH WHITE The basic aim of this note is to describe how to break rational functions into pieces. For example 2x + 3 = + x 3 x +. The point is that we don

More information

MEMORIAL UNIVERSITY OF NEWFOUNDLAND

MEMORIAL UNIVERSITY OF NEWFOUNDLAND MEMORIAL UNIVERSITY OF NEWFOUNDLAND DEPARTMENT OF MATHEMATICS AND STATISTICS Section 5. Math 090 Fall 009 SOLUTIONS. a) Using long division of polynomials, we have x + x x x + ) x 4 4x + x + 0x x 4 6x

More information

, a 1. , a 2. ,..., a n

, 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 information

Section 4.3. Polynomial Division; The Remainder Theorem and the Factor Theorem

Section 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 information

Dividing Polynomials: Remainder and Factor Theorems

Dividing 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 information

Chapter 3: Polynomial and Rational Functions

Chapter 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 information

Roots and Coefficients Polynomials Preliminary Maths Extension 1

Roots and Coefficients Polynomials Preliminary Maths Extension 1 Preliminary Maths Extension Question If, and are the roots of x 5x x 0, find the following. (d) (e) Question If p, q and r are the roots of x x x 4 0, evaluate the following. pq r pq qr rp p q q r r p

More information

The absolute value (modulus) of a number

The absolute value (modulus) of a number The absolute value (modulus) of a number Given a real number x, its absolute value or modulus is dened as x if x is positive x = 0 if x = 0 x if x is negative For example, 6 = 6, 10 = ( 10) = 10. The absolute

More information

Student: 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 information

Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).

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

More information

Skills Practice Skills Practice for Lesson 10.1

Skills 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 information

Maths Extension 2 - Polynomials. Polynomials

Maths Extension 2 - Polynomials. Polynomials Maths Extension - Polynomials Polynomials! Definitions and properties of polynomials! Factors & Roots! Fields ~ Q Rational ~ R Real ~ C Complex! Finding zeros over the complex field! Factorization & Division

More information

2 the maximum/minimum value is ( ).

2 the maximum/minimum value is ( ). Math 60 Ch3 practice Test The graph of f(x) = 3(x 5) + 3 is with its vertex at ( maximum/minimum value is ( ). ) and the The graph of a quadratic function f(x) = x + x 1 is with its vertex at ( the maximum/minimum

More information

More Polynomial Equations Section 6.4

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

More information

Section 3.1: Characteristics of Polynomial Functions

Section 3.1: Characteristics of Polynomial Functions Chapter 3: Polynomial Functions Section 3.1: Characteristics of Polynomial Functions pg 107 Polynomial Function: a function of the form f(x) = a n x n + a n 1 x n 1 +a n 2 x n 2 +...+a 2 x 2 +a 1 x+a 0

More information

L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen

L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to

More information

Package Summary. Linear Equations Quadratic Equations Rational Equations. Making Math Possible 1 of 10 c Sa diyya Hendrickson

Package Summary. Linear Equations Quadratic Equations Rational Equations. Making Math Possible 1 of 10 c Sa diyya Hendrickson Solving Equations Prepared by: Sa diyya Hendrickson Name: Date: Package Summary Linear Equations Quadratic Equations Rational Equations Making Math Possible 1 of 10 c Sa diyya Hendrickson Linear Equations

More information

L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen

L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to

More information

General Form: y = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0

General Form: y = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 Families of Functions Prepared by: Sa diyya Hendrickson Name: Date: Definition: function A function f is a rule that relates two sets by assigning to some element (e.g. x) in a set A exactly one element

More information

ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS

ZEROS 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 information

8.4 Partial Fractions

8.4 Partial Fractions 8.4 1 8.4 Partial Fractions Consider the following integral. (1) 13 2x x 2 x 2 dx How might we evaluate this? Suppose that, by some good luck, we knew that (2) 13 2x x 2 x 2 = 3 x 2 5 x+1 We could then

More information

Polynomial Functions and Models

Polynomial Functions and Models 1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models

More information

Partial Fractions. Calculus 2 Lia Vas

Partial Fractions. Calculus 2 Lia Vas Calculus Lia Vas Partial Fractions rational function is a quotient of two polynomial functions The method of partial fractions is a general method for evaluating integrals of rational function The idea

More information

Ch. 12 Higher Degree Equations Rational Root

Ch. 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 information

Tenth Maths Polynomials

Tenth Maths Polynomials Tenth Maths Polynomials Polynomials are algebraic expressions constructed using constants and variables. Coefficients operate on variables, which can be raised to various powers of non-negative integer

More information

3.5. Dividing Polynomials. LEARN ABOUT the Math. Selecting a strategy to divide a polynomial by a binomial

3.5. Dividing Polynomials. LEARN ABOUT the Math. Selecting a strategy to divide a polynomial by a binomial 3.5 Dividing Polynomials GOAL Use a variety of strategies to determine the quotient when one polynomial is divided by another polynomial. LEARN ABOU the Math Recall that long division can be used to determine

More information

Integration of Rational Functions by Partial Fractions

Integration of Rational Functions by Partial Fractions Title Integration of Rational Functions by MATH 1700 MATH 1700 1 / 11 Readings Readings Readings: Section 7.4 MATH 1700 2 / 11 Rational functions A rational function is one of the form where P and Q are

More information

Chapter 2.7 and 7.3. Lecture 5

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

More information

Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:

Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example: Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: 10 + 5x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a

More information

Chapter 2: Polynomial and Rational Functions

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.

More information

Polynomial expression

Polynomial expression 1 Polynomial expression Polynomial expression A expression S(x) in one variable x is an algebraic expression in x term as Where an,an-1,,a,a0 are constant and real numbers and an is not equal to zero Some

More information

Integration of Rational Functions by Partial Fractions

Integration of Rational Functions by Partial Fractions Title Integration of Rational Functions by Partial Fractions MATH 1700 December 6, 2016 MATH 1700 Partial Fractions December 6, 2016 1 / 11 Readings Readings Readings: Section 7.4 MATH 1700 Partial Fractions

More information

3.3 Dividing Polynomials. Copyright Cengage Learning. All rights reserved.

3.3 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 information

171S4.3 Polynomial Division; The Remainder and Factor Theorems. October 26, Polynomial Division; The Remainder and Factor Theorems

171S4.3 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 information

171S4.3 Polynomial Division; The Remainder and Factor Theorems. March 24, Polynomial Division; The Remainder and Factor Theorems

171S4.3 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 information

MATH 103 Pre-Calculus Mathematics Test #3 Fall 2008 Dr. McCloskey Sample Solutions

MATH 103 Pre-Calculus Mathematics Test #3 Fall 2008 Dr. McCloskey Sample Solutions MATH 103 Pre-Calculus Mathematics Test #3 Fall 008 Dr. McCloskey Sample Solutions 1. Let P (x) = 3x 4 + x 3 x + and D(x) = x + x 1. Find polynomials Q(x) and R(x) such that P (x) = Q(x) D(x) + R(x). (That

More information

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.

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

More information

Algebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions

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

More information

6.4 Division of Polynomials. (Long Division and Synthetic Division)

6.4 Division of Polynomials. (Long Division and Synthetic Division) 6.4 Division of Polynomials (Long Division and Synthetic Division) When we combine fractions that have a common denominator, we just add or subtract the numerators and then keep the common denominator

More information

Math123 Lecture 1. Dr. Robert C. Busby. Lecturer: Office: Korman 266 Phone :

Math123 Lecture 1. Dr. Robert C. Busby. Lecturer: Office: Korman 266 Phone : Lecturer: Math1 Lecture 1 Dr. Robert C. Busby Office: Korman 66 Phone : 15-895-1957 Email: rbusby@mcs.drexel.edu Course Web Site: http://www.mcs.drexel.edu/classes/calculus/math1_spring0/ (Links are case

More information

Solving Quadratic and Other Polynomial Equations

Solving Quadratic and Other Polynomial Equations Section 4.3 Solving Quadratic and Other Polynomial Equations TERMINOLOGY 4.3 Previously Used: This is a review of terms with which you should already be familiar. Formula New Terms to Learn: Discriminant

More information

Ch 7 Summary - POLYNOMIAL FUNCTIONS

Ch 7 Summary - POLYNOMIAL FUNCTIONS Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)

More information

S56 (5.1) Polynomials.notebook August 25, 2016

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,

More information

3 Polynomial and Rational Functions

3 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 information

MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions

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,

More information

Factors, Zeros, and Roots

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.

More information

Unit 1: Polynomial Functions SuggestedTime:14 hours

Unit 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 information

Answers. 2. List all theoretically possible rational roots of the polynomial: P(x) = 2x + 3x + 10x + 14x ) = A( x 4 + 3x 2 4)

Answers. 2. List all theoretically possible rational roots of the polynomial: P(x) = 2x + 3x + 10x + 14x ) = A( x 4 + 3x 2 4) CHAPTER 5 QUIZ Tuesday, April 1, 008 Answers 5 4 1. P(x) = x + x + 10x + 14x 5 a. The degree of polynomial P is 5 and P must have 5 zeros (roots). b. The y-intercept of the graph of P is (0, 5). The number

More information

Chapter Five Notes N P U2C5

Chapter 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 information

Advanced Math Quiz Review Name: Dec Use Synthetic Division to divide the first polynomial by the second polynomial.

Advanced Math Quiz Review Name: Dec Use Synthetic Division to divide the first polynomial by the second polynomial. Advanced Math Quiz 3.1-3.2 Review Name: Dec. 2014 Use Synthetic Division to divide the first polynomial by the second polynomial. 1. 5x 3 + 6x 2 8 x + 1, x 5 1. Quotient: 2. x 5 10x 3 + 5 x 1, x + 4 2.

More information

b n x n + b n 1 x n b 1 x + b 0

b n x n + b n 1 x n b 1 x + b 0 Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)

More information

Equations in Quadratic Form

Equations in Quadratic Form Equations in Quadratic Form MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: make substitutions that allow equations to be written

More information

Homework 8 Solutions to Selected Problems

Homework 8 Solutions to Selected Problems Homework 8 Solutions to Selected Problems June 7, 01 1 Chapter 17, Problem Let f(x D[x] and suppose f(x is reducible in D[x]. That is, there exist polynomials g(x and h(x in D[x] such that g(x and h(x

More information

Polynomial and Synthetic Division

Polynomial 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 information

Prepared by Sa diyya Hendrickson. Package Summary

Prepared by Sa diyya Hendrickson. Package Summary Introduction Prepared by Sa diyya Hendrickson Name: Date: Figure 1: c Cengage Learning Package Summary Definition and Properties of Exponents Understanding Properties (Frayer Models) Discovering Zero and

More information

+ 1 3 x2 2x x3 + 3x 2 + 0x x x2 2x + 3 4

+ 1 3 x2 2x x3 + 3x 2 + 0x x x2 2x + 3 4 Math 4030-001/Foundations of Algebra/Fall 2017 Polynomials at the Foundations: Rational Coefficients The rational numbers are our first field, meaning that all the laws of arithmetic hold, every number

More information

POLYNOMIALS. x + 1 x x 4 + x 3. x x 3 x 2. x x 2 + x. x + 1 x 1

POLYNOMIALS. x + 1 x x 4 + x 3. x x 3 x 2. x x 2 + x. x + 1 x 1 POLYNOMIALS A polynomial in x is an expression of the form p(x) = a 0 + a 1 x + a x +. + a n x n Where a 0, a 1, a. a n are real numbers and n is a non-negative integer and a n 0. A polynomial having only

More information

7.4: Integration of rational functions

7.4: Integration of rational functions A rational function is a function of the form: f (x) = P(x) Q(x), where P(x) and Q(x) are polynomials in x. P(x) = a n x n + a n 1 x n 1 + + a 0. Q(x) = b m x m + b m 1 x m 1 + + b 0. How to express a

More information

Power and Polynomial Functions. College Algebra

Power 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 information

Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations

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

More information

POLYNOMIALS. Maths 4 th ESO José Jaime Noguera

POLYNOMIALS. Maths 4 th ESO José Jaime Noguera POLYNOMIALS Maths 4 th ESO José Jaime Noguera 1 Algebraic expressions Book, page 26 YOUR TURN: exercises 1, 2, 3. Exercise: Find the numerical value of the algebraic expression xy 2 8x + y, knowing that

More information

Section 3.1 Quadratic Functions

Section 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 information

We say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:

We 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 information

Math-3. Lesson 3-1 Finding Zeroes of NOT nice 3rd Degree Polynomials

Math-3. Lesson 3-1 Finding Zeroes of NOT nice 3rd Degree Polynomials Math- Lesson - Finding Zeroes of NOT nice rd Degree Polynomials f ( ) 4 5 8 Is this one of the nice rd degree polynomials? a) Sum or difference of two cubes: y 8 5 y 7 b) rd degree with no constant term.

More information

1) Synthetic Division: The Process. (Ruffini's rule) 2) Remainder Theorem 3) Factor Theorem

1) 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 information

Mathematics 136 Calculus 2 Everything You Need Or Want To Know About Partial Fractions (and maybe more!) October 19 and 21, 2016

Mathematics 136 Calculus 2 Everything You Need Or Want To Know About Partial Fractions (and maybe more!) October 19 and 21, 2016 Mathematics 36 Calculus 2 Everything You Need Or Want To Know About Partial Fractions (and maybe more!) October 9 and 2, 206 Every rational function (quotient of polynomials) can be written as a polynomial

More information

Use the Rational Zero Theorem to list all the possible rational zeros of the following polynomials. (1-2) 4 3 2

Use the Rational Zero Theorem to list all the possible rational zeros of the following polynomials. (1-2) 4 3 2 Name: Math 114 Activity 1(Due by EOC Apr. 17) Dear Instructor or Tutor, These problems are designed to let my students show me what they have learned and what they are capable of doing on their own. Please

More information

Honours Advanced Algebra Unit 2: Polynomial Functions Factors, Zeros, and Roots: Oh My! Learning Task (Task 5) Date: Period:

Honours Advanced Algebra Unit 2: Polynomial Functions Factors, Zeros, and Roots: Oh My! Learning Task (Task 5) Date: Period: Honours Advanced Algebra Name: Unit : Polynomial Functions Factors, Zeros, and Roots: Oh My! Learning Task (Task 5) Date: Period: Mathematical Goals Know and apply the Remainder Theorem Know and apply

More information

ADDITONAL MATHEMATICS

ADDITONAL MATHEMATICS ADDITONAL MATHEMATICS 2002 2011 CLASSIFIED REMAINDER THEOREM Compiled & Edited By Dr. Eltayeb Abdul Rhman www.drtayeb.tk First Edition 2011 7 5 (i) Show that 2x 1 is a factor of 2x 3 5x 2 + 10x 4. [2]

More information

Lesson 19 Factoring Polynomials

Lesson 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 information

MATHEMATICAL METHODS UNIT 1 CHAPTER 4 CUBIC POLYNOMIALS

MATHEMATICAL METHODS UNIT 1 CHAPTER 4 CUBIC POLYNOMIALS E da = q ε ( B da = 0 E ds = dφ. B ds = μ ( i + μ ( ε ( dφ 3 MATHEMATICAL METHODS UNIT 1 CHAPTER 4 CUBIC POLYNOMIALS dt dt Key knowledge The key features and properties of cubic polynomials functions and

More information

The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts.

The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts. Math 141 Review for Final The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts. Part 1 (no calculator) graphing (polynomial, rational, linear, exponential, and logarithmic

More information

Unit 2 Rational Functionals Exercises MHF 4UI Page 1

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. *)

More information

Polynomial Review Problems

Polynomial Review Problems Polynomial Review Problems 1. Find polynomial function formulas that could fit each of these graphs. Remember that you will need to determine the value of the leading coefficient. The point (0,-3) is on

More information

Polynomial and Synthetic Division

Polynomial and Synthetic Division Chapter Polynomial Functions. f y. Common function: y Transformation: Vertical stretch each y-value is multiplied by, then a vertical shift nine units upward f Horizontal shift of three units to the left,

More information

Theorems About Roots of Polynomial Equations. Theorem Rational Root Theorem

Theorems About Roots of Polynomial Equations. Theorem Rational Root Theorem - Theorems About Roots of Polynomial Equations Content Standards N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. Also N.CN.8 Objectives To solve equations using the

More information

10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions

10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions Lesson 15 Graphs of Rational Functions SKILLS REVIEW! Use function composition to prove that the following two funtions are inverses of each other. 2x 3 f(x) = g(x) = 5 2 x 1 1 2 Lesson Objectives! The

More information

Polynomial Functions

Polynomial 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 information

MHF4U Unit 2 Polynomial Equation and Inequalities

MHF4U Unit 2 Polynomial Equation and Inequalities MHF4U Unit 2 Polynomial Equation and Inequalities Section Pages Questions Prereq Skills 82-83 # 1ac, 2ace, 3adf, 4, 5, 6ace, 7ac, 8ace, 9ac 2.1 91 93 #1, 2, 3bdf, 4ac, 5, 6, 7ab, 8c, 9ad, 10, 12, 15a,

More information

Table of contents. Polynomials Quadratic Functions Polynomials Graphs of Polynomials Polynomial Division Finding Roots of Polynomials

Table 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 information

Two hours. To be provided by Examinations Office: Mathematical Formula Tables. THE UNIVERSITY OF MANCHESTER. 29 May :45 11:45

Two hours. To be provided by Examinations Office: Mathematical Formula Tables. THE UNIVERSITY OF MANCHESTER. 29 May :45 11:45 Two hours MATH20602 To be provided by Examinations Office: Mathematical Formula Tables. THE UNIVERSITY OF MANCHESTER NUMERICAL ANALYSIS 1 29 May 2015 9:45 11:45 Answer THREE of the FOUR questions. If more

More information

3 What is the degree of the polynomial function that generates the data shown below?

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

More information

3.4. ZEROS OF POLYNOMIAL FUNCTIONS

3.4. 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 information

P4 Polynomials and P5 Factoring Polynomials

P4 Polynomials and P5 Factoring Polynomials P4 Polynomials and P5 Factoring Polynomials Professor Tim Busken Graduate T.A. Dynamical Systems Program Department of Mathematics San Diego State University June 22, 2011 Professor Tim Busken (Graduate

More information