Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
|
|
- Joseph Ramsey
- 7 years ago
- Views:
Transcription
1 1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c 2 x n c n 1 x + c n where n is an integer 0, and c 0, c 1,..., c n are constants. For example 2x 6 + 5x 5 + x x2 6x 10 Note that x and 4x are not polynomials. x 2 The constants c 0,..., c n are called the coefficients. The constant term is c n. The leading term is the term involving the highest power of x, here c 0 x n. The degree is the power of x in the leading term. A degree 0 polynomial is just a constant, e.g., 2 is (a) constant A degree 1 polynomial is called linear, e.g., 3x + 2 is linear A degree 2 polynomial is called quadratic, e.g., x 2 + 2x + 1 is quadratic A degree 3 polynomial is called cubic, e.g., y 3 + 7y 2 is a cubic in y Operations Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if then f(x) = x 2 3, g(x) = 4x 3 + 3x 2 2x + 6 f(x) + g(x) = 4x 3 + 4x 2 2x + 3, f(x) g(x) = 4x 3 2x 2 + 2x 9 To multiply two polynomials, every term of the first polynomial must be multiplied by every term of the second. Example 1.1. f(x)g(x) =(x 2 3)(4x 3 + 3x 2 2x + 6) =x 2 (4x 3 + 3x 2 2x + 6) 3(4x 3 + 3x 2 2x + 6) =4x 5 + 3x 4 2x 3 + 6x 2 12x 3 9x 2 + 6x 18 =4x 5 + 3x 4 14x 3 3x 2 + 6x 18 1
2 1.3. Polynomial division Similar to long division for ordinary numbers. To divide 2x x 2 3x + 1 by x + 3, we compute 2x 2 +4x 15 x + 3 2x x 2 3x +1 2x 3 + 6x 2 4x 2 3x 4x 2 +12x 15x +1 15x The term 2x 3 must come from multiplying x+3 by 2x 2, so put 2x 2 on the top line, multiply x + 3 by 2x 2 and subtract from the line above. Now bring down the 3x. Multiply x + 3 by 4x and subtract. Bring down the 1. Multiply x + 3 by 15 and subtract. This completes the process. The quotient is 2x 2 + 4x 15 and the remainder is 46. polynomial = divisor quotient + remainder 2x x 2 3x + 1 = (x + 3)(2x 2 + 4x 15) + 46 Note. If the remainder is zero, we say that the polynomial is exactly divisible by the divisor, or that the divisor is a factor of the polynomial. Example 1.2. Divide x 4 2x 2 + 3x 6 by x 2 4x + 3. x 2 +4x +11 x 2 4x + 3 x 4 2x 2 +3x 6 x 4 4x 3 +3x 2 4x 3 5x 2 +3x 4x 3 16x 2 +12x 11x 2 9x 6 11x 2 44x x 39 The quotient is x 2 + 4x The remainder is 35x 39. polynomial = divisor quotient + remainder x 4 2x 2 + 3x 6 = (x 2 4x + 3)(x 2 + 4x + 11) + 35x 39 Note. It is always true that degree of remainder < degree of divisor. 2
3 1.4. Remainder Theorem Theorem 1.3. When a polynomial f(x) is divided by x a, then the remainder is f(a). Example 1.4. The remainder when f(x) = 2x 5 3x is divided by x + 2 is f( 2) = ( 2) 5 3( 2) = = 43. Check this yourself by doing the long division. Proof. If the quotient is q(x) and the remainder is r, then f(x) = (x a).q(x) + r Substituting x = a, gives f(a) = (a a) q(a) + r = r Factor Theorem Theorem 1.5. If x a is a factor of a polynomial f(x), then f(a) = 0. Conversely, if f(a) = 0, then x a is a factor of f(x). Example 1.6. Is x 1 is a factor of f(x) = x 5 1? Since f(1) = 0, the Factor Theorem tells us it must be. In fact x 5 1 = (x 1)(x 4 + x 3 + x 2 + x + 1). Proof. If x a is a factor of f(x) then f(x) = (x a) g(x) for some polynomial g(x). Therefore f(a) = (a a) g(a) = 0. If f(a) = 0 then by the Remainder Theorem, the remainder on dividing f(x) by x a is 0. Therefore f(x) = (x a) q(x) where q(x) is the quotient Factorizing polynomials It is often useful to write a polynomial as a product of polynomials of lower degree. For example f(x) = x 3 2x 2 x + 2 can be factorized as (x 2 1)(x 2) or as (x 1)(x + 1)(x 2). Factorizing helps find the roots since, if a product of numbers is zero, then one of them must be zero. Therefore if f(x) = 0 then x 1 = 0, or x + 1 = 0 or x 2 = 0. Therefore x = 1, 1 or 2. Some well-known factorizations: (i) Difference of squares: x 2 a 2 = (x a)(x + a) for example, x 2 9 = (x + 3)(x 3). (ii) Perfect square: x 2 ± 2ax + a 2 = (x ± a) 2 for example x 2 10x + 25 = (x 5) 2. (iii) If there is no constant term, then x is a common factor. For example, x 2 + 4x = x(x + 4). 3
4 1.7. Factorizing by trial and error Given a quadratic x 2 + bx + c, we try to write it as x 2 + bx + c = (x + r)(x + s) for some r, s. Comparing coefficients, we need to find two numbers r and s with r + s = b and rs = c. If r and s are integers, they must be factors of c. Sometimes they can be found by trial and error. Example 1.7. x 2 + 7x Need rs = 10. You can write 10 = 1 10 = 2 5. The second one works since = 7. Therefore x 2 + 7x + 10 = (x + 2)(x + 5). Example 1.8. x 2 + 3x 18. Need rs = 18, so one of the numbers r and s must be positive, the other negative. Now 18 = 1 18 = 2 9 = 3 6. In fact r = 6, s = 3 works, since r + s = 3. Therefore x 2 + 3x 18 = (x + 6)(x 3). To factorize ax 2 + bx + c with a 1 by inspection is harder, but it can sometimes be done. Example x 2 + 7x + 6. Try to write it as (2x + r)(x + s). Need rs = 6 and r + 2s = 7. Take r = 3 and s = 2, so 2x 2 + 7x + 6 = (2x + 3)(x + 2) Using the Factor Theorem If f(a) = 0 then x a is a factor of f(x). If f(x) = ax 2 + bx + c has roots x = x 1, x 2, then f(x) = a(x x 1 )(x x 2 ). Example From the standard formula, x 2 + 6x 72 has roots x = 6 ± 324 giving 2 x = 6, 12. Therefore x 2 + 6x 72 = (x 6)(x + 12). It can be difficult to factorize polynomials of degree 3. If you happen to know a root, say f(a) = 0, then x a is a factor, so f(x) = (x a) q(x). Use polynomial division to find q(x), and try to factorize it Worked examples Example Find the quotient and remainder for (x 3 + 5x 2 + 4x 17) (x 4). Example Find the quotient and remainder for (x 3 4x 2 + x + 2) (x 2 3). Example When x 3 4x 2 + 5x + a is divided by x 3, the remainder is 7. Find a. Example Factorize the following polynomials: x 2 + 7x + 12, x 2 + 8x, x 3 5x 2 50x, 2x 2 5x + 3, x 3 + 3x 2 4x 12. Example The polynomial x 3 + 8x 2 + kx + 10 has a factor x + 2. (a) Find the value of k. (b) Factorize the polynomial. 4
5 Example Given that f(x) = x 3 4x 2 x+4, find f(1) and f(2), and hence factorize f(x) into a product of three linear factors. Example Express 6x x x + 3 in the form (x + 1)(ax 2 + bx + c), stating the values of a, b and c, and hence solve the equation 6x x x + 3 = 0. 5
JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationSOLVING POLYNOMIAL EQUATIONS
C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra
More informationLagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.
Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More information1 Shapes of Cubic Functions
MA 1165 - Lecture 05 1 1/26/09 1 Shapes of Cubic Functions A cubic function (a.k.a. a third-degree polynomial function) is one that can be written in the form f(x) = ax 3 + bx 2 + cx + d. (1) Quadratic
More information3.2 The Factor Theorem and The Remainder Theorem
3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More information1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).
.7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More information3.6. The factor theorem
3.6. The factor theorem Example 1. At the right we have drawn the graph of the polynomial y = x 4 9x 3 + 8x 36x + 16. Your problem is to write the polynomial in factored form. Does the geometry of the
More informationApplication. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.
Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: 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 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More information2.5 Zeros of a Polynomial Functions
.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and
More information3.6 The Real Zeros of a Polynomial Function
SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,
More informationThe Division Algorithm for Polynomials Handout Monday March 5, 2012
The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationSect 6.7 - Solving Equations Using the Zero Product Rule
Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred
More informationFactoring Polynomials
Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}
More informationThe Factor Theorem and a corollary of the Fundamental Theorem of Algebra
Math 421 Fall 2010 The Factor Theorem and a corollary of the Fundamental Theorem of Algebra 27 August 2010 Copyright 2006 2010 by Murray Eisenberg. All rights reserved. Prerequisites Mathematica Aside
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationFACTORING QUADRATIC EQUATIONS
FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationVieta s Formulas and the Identity Theorem
Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion
More informationSolving Quadratic Equations by Factoring
4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written
More informationFactoring Polynomials
Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations
More informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
More information63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.
9.4 (9-27) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27-in. wheel, 44 teeth
More informationModern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)
Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)
More information2.4 Real Zeros of Polynomial Functions
SECTION 2.4 Real Zeros of Polynomial Functions 197 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower
More information9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.
9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationPolynomial Expressions and Equations
Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen
More informationis identically equal to x 2 +3x +2
Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3
More informationCollege Algebra - MAT 161 Page: 1 Copyright 2009 Killoran
College Algebra - MAT 6 Page: Copyright 2009 Killoran Zeros and Roots of Polynomial Functions Finding a Root (zero or x-intercept) of a polynomial is identical to the process of factoring a polynomial.
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationPartial Fractions Examples
Partial Fractions Examples Partial fractions is the name given to a technique of integration that may be used to integrate any ratio of polynomials. A ratio of polynomials is called a rational function.
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More information8 Polynomials Worksheet
8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationCHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationUNCORRECTED PAGE PROOFS
number and and algebra TopIC 17 Polynomials 17.1 Overview Why learn this? Just as number is learned in stages, so too are graphs. You have been building your knowledge of graphs and functions over time.
More information0.4 FACTORING POLYNOMIALS
36_.qxd /3/5 :9 AM Page -9 SECTION. Factoring Polynomials -9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationTim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date
Leaving Certificate Honours Maths - Algebra the date Chapter 1 Algebra This is an important portion of the course. As well as generally accounting for 2 3 questions in examination it is the basis for many
More information5. Factoring by the QF method
5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the
More informationPROBLEM SET 6: POLYNOMIALS
PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other
More informationPolynomials and Factoring
Lesson 2 Polynomials and Factoring A polynomial function is a power function or the sum of two or more power functions, each of which has a nonnegative integer power. Because polynomial functions are built
More informationSome facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)
Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of the
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationFactorising quadratics
Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to
More informationPolynomial and Rational Functions
Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving
More information3-17 15-25 5 15-10 25 3-2 5 0. 1b) 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
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationAlgebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials
More informationBasics of Polynomial Theory
3 Basics of Polynomial Theory 3.1 Polynomial Equations In geodesy and geoinformatics, most observations are related to unknowns parameters through equations of algebraic (polynomial) type. In cases where
More informationWinter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com
Polynomials Alexander Remorov alexanderrem@gmail.com Warm-up Problem 1: Let f(x) be a quadratic polynomial. Prove that there exist quadratic polynomials g(x) and h(x) such that f(x)f(x + 1) = g(h(x)).
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationTool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationFactoring Polynomials
Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,
More informationUnique Factorization
Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon
More informationReal Roots of Univariate Polynomials with Real Coefficients
Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More informationDeterminants can be used to solve a linear system of equations using Cramer s Rule.
2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution
More informationFOIL FACTORING. Factoring is merely undoing the FOIL method. Let s look at an example: Take the polynomial x²+4x+4.
FOIL FACTORING Factoring is merely undoing the FOIL method. Let s look at an example: Take the polynomial x²+4x+4. First we take the 3 rd term (in this case 4) and find the factors of it. 4=1x4 4=2x2 Now
More informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More information3.3. The Factor Theorem. Investigate Determining the Factors of a Polynomial. Reflect and Respond
3.3 The Factor Theorem Focus on... factoring polynomials explaining the relationship between the linear factors of a polynomial expression and the zeros of the corresponding function modelling and solving
More informationis identically equal to x 2 +3x +2
Partial fractions.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as + for any
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationFactoring Polynomials
Factoring Polynomials 4-1-2014 The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial? Let p(x) be a polynomial. p(c) = 0 is equivalent to x c dividing p(x). Recall
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
MODULE - 1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write
More information3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes
Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same
More informationMarch 29, 2011. 171S4.4 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 informationALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section
ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More informationAlgebra 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
More informationFactoring Polynomials
Factoring Polynomials Any Any Any natural number that that that greater greater than than than 1 1can can 1 be can be be factored into into into a a a product of of of prime prime numbers. For For For
More informationSolving Cubic Polynomials
Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial
More information3 1. Note that all cubes solve it; therefore, there are no more
Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if
More information3.3 Real Zeros of Polynomials
3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.
More information7. Some irreducible polynomials
7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationDefinitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).
Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32
More informationMore Quadratic Equations
More Quadratic Equations Math 99 N1 Chapter 8 1 Quadratic Equations We won t discuss quadratic inequalities. Quadratic equations are equations where the unknown appears raised to second power, and, possibly
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More information