SOLVING POLYNOMIAL EQUATIONS

Save this PDF as:

Size: px
Start display at page:

Transcription

1 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 book. A BRIEF REVIEW OF POLYNOMIALS Recall that if n is a nonnegative integer, then a polynomial of degree n is a function that can be written in the following forms, depending on whether you want the powers of x in ascending or descending order: c 0 + c 1 x + c 2 x 2 + +c n x n (c n = 0) c n x n + c n 1 x n 1 + +c 1 x + c 0 (c n = 0) The numbers c 0,c 1,...,c n are called the coefficients of the polynomial. The coefficient c n (which multiplies the highest power of x) is called the leading coefficient, the term c n x n is called the leading term, and the coefficient c 0 is called the constant term. Polynomials of degree 1, 2, 3, 4, and 5 are called linear, quadratic, cubic, quartic, and quintic, respectively. For simplicity, general polynomials of low degree are often written without subscripts on the coefficients: p(x) = a Constant polynomial p(x) = ax + b (a = 0) Linear polynomial p(x) = ax 2 + bx + c (a = 0) Quadratic polynomial p(x) = ax 3 + bx 2 + cx + d (a = 0) Cubic polynomial When you attempt to factor a polynomial completely, one of three things can happen: You may be able to decompose the polynomial into distinct linear factors using only real numbers; for example, x 3 + x 2 2x = x(x 2 + x 2) = x(x 1)(x + 2) You may be able to decompose the polynomial into linear factors using only real numbers, but some of the factors may be repeated; for example, x 6 3x 4 + 2x 3 = x 3 (x 3 3x + 2) = x 3 (x 1) 2 (x + 2) (1) You may be able to decompose the polynomial into linear and quadratic factors using only real numbers, but you may not be able to decompose the quadratic factors into linear factors using only real numbers (such quadratic factors are said to be irreducible over the real numbers); for example, x 4 1 = (x 2 1)(x 2 + 1) = (x 1)(x + 1)(x 2 + 1) = (x 1)(x + 1)(x i)(x + i) Here, the factor x is irreducible over the real numbers. C1

2 C2 Appendix C: Solving Polynomial Equations In general, if p(x) is a polynomial of degree n with leading coefficient a, and if complex numbers are allowed, then p(x) can be factored as p(x) = a(x r 1 )(x r 2 ) (x r n ) (2) where r 1,r 2,...,r n are called the zeros of p(x) or the roots of the equation p(x) = 0, and (2) is called the complete linear factorization of p(x). If some of the factors in (2) are repeated, then they can be combined; for example, if the first k factors are distinct and the rest are repetitions of the first k, then (2) can be expressed in the form p(x) = a(x r 1 ) m 1 (x r 2 ) m2 (x r k ) m k (3) where r 1,r 2,...,r k are the distinct roots of p(x) = 0. The exponents m 1,m 2,...,m k tell us how many times the various factors occur in the complete linear factorization; for example, in (3) the factor (x r 1 ) occurs m 1 times, the factor (x r 2 ) occurs m 2 times, and so forth. Some techniques for factoring polynomials are discussed later in this appendix. In general, if a factor (x r) occurs m times in the complete linear factorization of a polynomial, then we say that r is a root or zero of multiplicity m, and if (x r) has no repetitions (i.e., r has multiplicity 1), then we say that r is a simple root or zero. For example, it follows from (1) that the equation x 6 3x 4 + 2x 3 = 0 can be expressed as x 3 (x 1) 2 (x + 2) = 0 (4) so this equation has three distinct roots a root x = 0 of multiplicity 3, a root x = 1 of multiplicity 2, and a simple root x = 2. Note that in (3) the multiplicities of the roots must add up to n, since p(x) has degree n; that is, m 1 + m 2 + +m k = n For example, in (4) the multiplicities add up to 6, which is the same as the degree of the polynomial. It follows from (2) that a polynomial of degree n can have at most n distinct roots; if all of the roots are simple, then there will be exactly n, but if some are repeated, then there will be fewer than n. However, when counting the roots of a polynomial, it is standard practice to count multiplicities, since that convention allows us to say that a polynomial of degree n has n roots. For example, from (1) the six roots of the polynomial p(x) = x 6 3x 4 + 2x 3 are r = 0, 0, 0, 1, 1, 2 In summary, we have the following important theorem. C.1 theorem If complex roots are allowed, and if roots are counted according to their multiplicities, then a polynomial of degree n has exactly n roots. THE REMAINDER THEOREM When two positive integers are divided, the numerator can be expressed as the quotient plus the remainder over the divisor, where the remainder is less than the divisor. For example, 17 5 = If we multiply this equation through by 5, we obtain 17 = which states that the numerator is the divisor times the quotient plus the remainder.

3 Appendix C: Solving Polynomial Equations C3 The following theorem, which we state without proof, is an analogous result for division of polynomials. C.2 theorem If p(x) and s(x) are polynomials, and if s(x) is not the zero polynomial, then p(x) can be expressed as p(x) = s(x)q(x) + r(x) where q(x) and r(x) are the quotient and remainder that result when p(x) is divided by s(x), and either r(x) is the zero polynomial or the degree of r(x) is less than the degree of s(x). In the special case where p(x) is divided by a first-degree polynomial of the form x c, the remainder must be some constant r, since it is either zero or has degree less than 1. Thus, Theorem C.2 implies that p(x) = (x c)q(x) + r and this in turn implies that p(c) = r. In summary, we have the following theorem. C.3 theorem (Remainder Theorem) If a polynomial p(x) is divided by x c, then the remainder is p(c). Example 1 According to the Remainder Theorem, the remainder on dividing p(x) = 2x 3 + 3x 2 4x 3 by x + 4 should be p( 4) = 2( 4) 3 + 3( 4) 2 4( 4) 3 = 67 Show that this is so. Solution. By long division 2x 2 5x + 16 x + 4 2x 3 + 3x 2 4x 3 2x 3 + 8x 2 5x 2 4x 5x 2 20x 16x 3 16x which shows that the remainder is 67. Alternative Solution. Because we are dividing by an expression of the form x c (where c = 4), we can use synthetic division rather than long division. The computations are which again shows that the remainder is 67.

4 C4 Appendix C: Solving Polynomial Equations THE FACTOR THEOREM To factor a polynomial p(x) is to write it as a product of lower-degree polynomials, called factors of p(x). For s(x) to be a factor of p(x) there must be no remainder when p(x) is divided by s(x). For example, if p(x) can be factored as p(x) = s(x)q(x) (5) then p(x) = q(x) (6) s(x) so dividing p(x) by s(x) produces a quotient q(x) with no remainder. Conversely, (6) implies (5), so s(x) is a factor of p(x) if there is no remainder when p(x) is divided by s(x). In the special case where x c is a factor of p(x), the polynomial p(x) can be expressed as p(x) = (x c)q(x) which implies that p(c) = 0. Conversely, if p(c) = 0, then the Remainder Theorem implies that x c is a factor of p(x), since the remainder is 0 when p(x) is divided by x c. These results are summarized in the following theorem. C.4 theorem (Factor Theorem) A polynomial p(x) has a factor x c if and only if p(c) = 0. It follows from this theorem that the statements below say the same thing in different ways: x c is a factor of p(x). p(c) = 0. c is a zero of p(x). c is a root of the equation p(x) = 0. c is a solution of the equation p(x) = 0. c is an x-intercept of y = p(x). Example 2 Confirm that x 1 is a factor of p(x) = x 3 3x 2 13x + 15 by dividing x 1 into p(x) and checking that the remainder is zero. Solution. By long division x 2 2x 15 x 1 x 3 3x 2 13x + 15 x 3 x 2 2x 2 13x 2x 2 + 2x 15x x which shows that the remainder is zero.

5 Appendix C: Solving Polynomial Equations C5 Alternative Solution. Because we are dividing by an expression of the form x c,we can use synthetic division rather than long division. The computations are which again confirms that the remainder is zero. USING ONE FACTOR TO FIND OTHER FACTORS If x c is a factor of p(x), and if q(x) = p(x)/(x c), then p(x) = (x c)q(x) (7) so that additional linear factors of p(x) can be obtained by factoring the quotient q(x). Example 3 Factor p(x) = x 3 3x 2 13x + 15 (8) completely into linear factors. Solution. We showed in Example 2 that x 1 is a factor of p(x) and we also showed that p(x)/(x 1) = x 2 2x 15. Thus, x 3 3x 2 13x + 15 = (x 1)(x 2 2x 15) Factoring x 2 2x 15 by inspection yields x 3 3x 2 13x + 15 = (x 1)(x 5)(x + 3) which is the complete linear factorization of p(x). METHODS FOR FINDING ROOTS A general quadratic equation ax 2 + bx + c = 0 can be solved by using the quadratic formula to express the solutions of the equation in terms of the coefficients. Versions of this formula were known since Babylonian times, and by the seventeenth century formulas had been obtained for solving general cubic and quartic equations. However, attempts to find formulas for the solutions of general fifth-degree equations and higher proved fruitless. The reason for this became clear in 1829 when the French mathematician Evariste Galois ( ) proved that it is impossible to express the solutions of a general fifth-degree equation or higher in terms of its coefficients using algebraic operations. Today, we have powerful computer programs for finding the zeros of specific polynomials. For example, it takes only seconds for a computer algebra system, such as Mathematica or Maple, to show that the zeros of the polynomial p(x) = 10x 4 23x 3 10x x + 6 (9) are x = 1, x = 1 5, x = 3, and x = 2 (10) 2 The algorithms that these programs use to find the integer and rational zeros of a polynomial, if any, are based on the following theorem, which is proved in advanced algebra courses.

6 C6 Appendix C: Solving Polynomial Equations C.5 theorem Suppose that p(x) = c n x n + c n 1 x n 1 + +c 1 x + c 0 is a polynomial with integer coefficients. (a) If r is an integer zero of p(x), then r must be a divisor of the constant term c 0. (b) If r = a/b is a rational zero of p(x) in which all common factors of a and b have been canceled, then a must be a divisor of the constant term c 0, and b must be a divisor of the leading coefficient c n. For example, in (9) the constant term is 6 (which has divisors ±1, ±2, ±3, and ±6) and the leading coefficient is 10 (which has divisors ±1, ±2, ±5, and ±10). Thus, the only possible integer zeros of p(x) are ±1, ±2, ±3, ±6 and the only possible noninteger rational zeros are ± 1 2, ± 1 5, ± 1 10, ± 2 5, ± 3 2, ± 3 5, ± 3 10, ± 6 5 Using a computer, it is a simple matter to evaluate p(x) at each of the numbers in these lists to show that its only rational zeros are the numbers in (10). Example 4 Solve the equation x 3 + 3x 2 7x 21 = 0. Solution. The solutions of the equation are the zeros of the polynomial p(x) = x 3 + 3x 2 7x 21 We will look for integer zeros first. All such zeros must divide the constant term, so the only possibilities are ±1, ±3, ±7, and ±21. Substituting these values into p(x) (or using the method of Exercise 6) shows that x = 3 is an integer zero. This tells us that x + 3is a factor of p(x) and that p(x) can be written as x 3 + 3x 2 7x 21 = (x + 3)q(x) where q(x) is the quotient that results when x 3 + 3x 2 7x 21 is divided by x + 3. We leave it for you to perform the division and show that q(x) = x 2 7; hence, x 3 + 3x 2 7x 21 = (x + 3)(x 2 7) = (x + 3)(x + 7)(x 7) which tells us that the solutions of the given equation are x = 3, x = , and x = EXERCISE SET C C CAS 1 2 Find the quotient q(x) and the remainder r(x) that result when p(x) is divided by s(x). 1. (a) p(x) = x 4 + 3x 3 5x + 10; s(x) = x 2 x + 2 (b) p(x) = 6x x 2 + 5; s(x) = 3x 2 1 (c) p(x) = x 5 + x 3 + 1; s(x) = x 2 + x 2. (a) p(x) = 2x 4 3x 3 + 5x 2 + 2x + 7; s(x)= x 2 x + 1 (b) p(x) = 2x 5 + 5x 4 4x 3 + 8x 2 + 1; s(x)= 2x 2 x + 1 (c) p(x) = 5x 6 + 4x 2 + 5; s(x) = x Use synthetic division to find the quotient q(x) and the remainder r(x) that result when p(x) is divided by s(x). 3. (a) p(x) = 3x 3 4x 1; s(x) = x 2 (b) p(x) = x 4 5x 2 + 4; s(x) = x + 5 (c) p(x) = x 5 1; s(x) = x 1 4. (a) p(x) = 2x 3 x 2 2x + 1; s(x) = x 1 (b) p(x) = 2x 4 + 3x 3 17x 2 27x 9; s(x) = x + 4 (c) p(x) = x 7 + 1; s(x) = x 1

7 Appendix C: Solving Polynomial Equations C7 5. Let p(x) = 2x 4 + x 3 3x 2 + x 4. Use synthetic division and the Remainder Theorem to find p(0), p(1), p( 3), and p(7). 6. Let p(x) be the polynomial in Example 4. Use synthetic division and the Remainder Theorem to evaluate p(x) at x =±1, ±3, ±7, and ± Let p(x) = x 3 + 4x 2 + x 6. Find a polynomial q(x) and a constant r such that (a) p(x) = (x 2)q(x) + r (b) p(x) = (x + 1)q(x) + r. 8. Let p(x) = x 5 1. Find a polynomial q(x) and a constant r such that (a) p(x) = (x + 1)q(x) + r (b) p(x) = (x 1)q(x) + r. 9. In each part, make a list of all possible candidates for the rational zeros of p(x). (a) p(x) = x 7 + 3x 3 x + 24 (b) p(x) = 3x 4 2x 2 + 7x 10 (c) p(x) = x Find all integer zeros of p(x) = x 6 + 5x 5 16x 4 15x 3 12x 2 38x Factor the polynomials completely. 11. p(x) = x 3 2x 2 x p(x) = 3x 3 + x 2 12x p(x) = x x x x p(x) = 2x 4 + x 3 + 3x 2 + 3x p(x) = x 5 + 4x 4 4x 3 34x 2 45x 18 C 16. For each of the factorizations that you obtained in Exercises 11 15, check your answer using a CAS Find all real solutions of the equations. 17. x 3 + 3x 2 + 4x + 12 = x 3 5x 2 10x + 3 = x x x 2 8x 8 = x 4 x 3 14x 2 5x + 6 = x 5 2x 4 6x 3 + 5x 2 + 8x + 12 = 0 C 22. For each of the equations you solved in Exercises 17 21, check your answer using a CAS. 23. Find all values of k for which x 1 is a factor of the polynomial p(x) = k 2 x 3 7kx Is x + 3 a factor of x ? Justify your answer. C 25. A 3 cm thick slice is cut from a cube, leaving a volume of 196 cm 3. Use a CAS to find the length of a side of the original cube. 26. (a) Show that there is no positive rational number that exceeds its cube by 1. (b) Does there exist a real number that exceeds its cube by 1? Justify your answer. 27. Use the Factor Theorem to show each of the following. (a) x y is a factor of x n y n for all positive integer values of n. (b) x + y is a factor of x n y n for all positive even integer values of n. (c) x + y is a factor of x n + y n for all positive odd integer values of n.

a 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

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

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

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

3.6 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,

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

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

3.2 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

An Insight into Division Algorithm, Remainder and Factor Theorem

An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the

6.4 The Remainder Theorem

6.4. THE REMAINDER THEOREM 6.3.2 Check the two divisions we performed in Problem 6.12 by multiplying the quotient by the divisor, then adding the remainder. 6.3.3 Find the quotient and remainder when x

6 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

Introduction to polynomials

Worksheet 4.5 Polynomials Section 1 Introduction to polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n, (n N) where p 0, p 1,..., p n are constants and x os

Zeros 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

Polynomials Classwork

Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th -degree polynomial function Def.: A polynomial function of degree n in the vaiable

Zeros of Polynomial Functions

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

SYNTHETIC DIVISION AND THE FACTOR THEOREM

628 (11 48) Chapter 11 Functions In this section Synthetic Division The Factor Theorem Solving Polynomial Equations 11.6 SYNTHETIC DIVISION AND THE FACTOR THEOREM In this section we study functions defined

1.7. 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

9. 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

Sect 3.2 Synthetic Division

94 Objective 1: Sect 3.2 Synthetic Division Division Algorithm Recall that when dividing two numbers, we can check our answer by the whole number (quotient) times the divisor plus the remainder. This should

PUTNAM 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

5.1 The Remainder and Factor Theorems; Synthetic Division

5.1 The Remainder and Factor Theorems; Synthetic Division In this section you will learn to: understand the definition of a zero of a polynomial function use long and synthetic division to divide polynomials

3.7 Complex Zeros; Fundamental Theorem of Algebra

SECTION.7 Complex Zeros; Fundamental Theorem of Algebra 2.7 Complex Zeros; Fundamental Theorem of Algebra PREPARING FOR THIS SECTION Before getting started, review the following: Complex Numbers (Appendix,

Solutions to Self-Test for Chapter 4 c4sts - p1

Solutions to Self-Test for Chapter 4 c4sts - p1 1. Graph a polynomial function. Label all intercepts and describe the end behavior. a. P(x) = x 4 2x 3 15x 2. (1) Domain = R, of course (since this is a

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

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

Worksheet 4.7. Polynomials. Section 1. Introduction to Polynomials. A polynomial is an expression of the form

Worksheet 4.7 Polynomials Section 1 Introduction to Polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n (n N) where p 0, p 1,..., p n are constants and x is a

Lagrange 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

CLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column.

SYNTHETIC DIVISION CLASS NOTES When factoring or evaluating polynomials we often find that it is convenient to divide a polynomial by a linear (first degree) binomial of the form x k where k is a real

Factoring 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

UNCORRECTED 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.

College 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.

Copy 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,...}

3 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

Polynomials and Factoring

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

x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent.

Rules of Exponents: If n > 0, m > 0 are positive integers and x, y are any real numbers, then: x m x n = x m+n x m x n = xm n, if m n (x m ) n = x mn (xy) n = x n y n ( x y ) n = xn y n 1 Can we make sense

Dividing Polynomials; Remainder and Factor Theorems

Dividing Polynomials; Remainder and Factor Theorems In this section we will learn how to divide polynomials, an important tool needed in factoring them. This will begin our algebraic study of polynomials.

In this lesson you will learn to find zeros of polynomial functions that are not factorable.

2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has

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

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

MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

AND RATIONAL FUNCTIONS. 2 Quadratic. Polynomials

UNIT 5 POLYNOMIAL In prior units of Core-Plus Mathematics, you developed understanding and skill in the use of linear, quadratic, and inverse variation functions. Those functions are members of two larger

Modern 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)

5-6 The Remainder and Factor Theorems

Use synthetic substitution to find f (4) and f ( 2) for each function. 1. f (x) = 2x 3 5x 2 x + 14 Divide the function by x 4. The remainder is 58. Therefore, f (4) = 58. Divide the function by x + 2.

Chapter 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

2.4 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

0.4 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

Zeros of Polynomial Functions

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

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

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

1 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.

Dividing Polynomials VOCABULARY

- Dividing Polynomials TEKS FOCUS TEKS ()(C) Determine the quotient of a polynomial of degree three and degree four when divided by a polynomial of degree one and of degree two. TEKS ()(A) Apply mathematics

it 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

2.5 ZEROS OF POLYNOMIAL FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions.

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

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

Section 3.2 Polynomial Functions and Their Graphs

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

3.2. The Remainder Theorem. Investigate Polynomial Division

3.2 The Remainder Theorem Focus on... describing the relationship between polynomial long division and synthetic division dividing polynomials by binomials of the form x - a using long division or synthetic

Algebra Nation MAFS Videos and Standards Alignment Algebra 2

Section 1, Video 1: Linear Equations in One Variable - Part 1 Section 1, Video 2: Linear Equations in One Variable - Part 2 Section 1, Video 3: Linear Equations and Inequalities in Two Variables Section

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

Quotient Rings of Polynomial Rings

Quotient Rings of Polynomial Rings 8-7-009 Let F be a field. is a field if and only if p(x) is irreducible. In this section, I ll look at quotient rings of polynomial rings. Let F be a field, and suppose

CHAPTER 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.

Solving 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

expression 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

Factoring 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

The 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,

Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

CHAPTER 4. Test Bank Exercises in. Exercise Set 4.1

Test Bank Exercises in CHAPTER 4 Exercise Set 4.1 1. Graph the quadratic function f(x) = x 2 2x 3. Indicate the vertex, axis of symmetry, minimum 2. Graph the quadratic function f(x) = x 2 2x. Indicate

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

7. 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

SOLVING POLYNOMIAL EQUATIONS. 0 = x 3 + ax 2 + bx + c = ( y 1 3 a) 3 ( 1 + a y 3 a) 2 ( 1 + b y. 3 a) + c = y 3 + y ( b 1 3 a2) + c ab c + 2

SOLVING POLYNOMIAL EQUATIONS FRANZ LEMMERMEYER 1 Cubic Equations Consider x 3 + ax + bx + c = 0 Using the same trick as above we can transform this into a cubic equation in which the coefficient of x vanishes:

Polynomial Expressions and Equations

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

Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials

Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial

PROBLEM 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

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an

3. Power of a Product: Separate letters, distribute to the exponents and the bases

Chapter 5 : Polynomials and Polynomial Functions 5.1 Properties of Exponents Rules: 1. Product of Powers: Add the exponents, base stays the same 2. Power of Power: Multiply exponents, bases stay the same

Cubic Functions: Global Analysis

Chapter 14 Cubic Functions: Global Analysis The Essential Question, 231 Concavity-sign, 232 Slope-sign, 234 Extremum, 235 Height-sign, 236 0-Concavity Location, 237 0-Slope Location, 239 Extremum Location,

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

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

The 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

Factoring Cubic Polynomials

Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than

The finite field with 2 elements The simplest finite field is

The finite field with 2 elements The simplest finite field is GF (2) = F 2 = {0, 1} = Z/2 It has addition and multiplication + and defined to be 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 0 0 = 0 0 1 = 0

3.4 Complex Zeros and the Fundamental Theorem of Algebra

86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

Polynomial 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

Partial Fractions. p(x) q(x)

Partial Fractions Introduction to Partial Fractions Given a rational function of the form p(x) q(x) where the degree of p(x) is less than the degree of q(x), the method of partial fractions seeks to break

SOLVING POLYNOMIAL EQUATIONS BY RADICALS Lee Si Ying 1 and Zhang De-Qi 2 1 Raffles Girls School (Secondary), 20 Anderson Road, Singapore 259978 2 Department of Mathematics, National University of Singapore,

The 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

calculating the result modulo 3, as follows: p(0) = 0 3 + 0 + 1 = 1 0,

Homework #02, due 1/27/10 = 9.4.1, 9.4.2, 9.4.5, 9.4.6, 9.4.7. Additional problems recommended for study: (9.4.3), 9.4.4, 9.4.9, 9.4.11, 9.4.13, (9.4.14), 9.4.17 9.4.1 Determine whether the following polynomials

ALGEBRA 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

Zeros of Polynomial Functions

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

MA107 Precalculus Algebra Exam 2 Review Solutions

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

An Introduction to Galois Fields and Reed-Solomon Coding

An Introduction to Galois Fields and Reed-Solomon Coding James Westall James Martin School of Computing Clemson University Clemson, SC 29634-1906 October 4, 2010 1 Fields A field is a set of elements on

UNIT 3: POLYNOMIALS AND ALGEBRAIC FRACTIONS. A polynomial is an algebraic expression that consists of a sum of several monomials. x n 1...

UNIT 3: POLYNOMIALS AND ALGEBRAIC FRACTIONS. Polynomials: A polynomial is an algebraic expression that consists of a sum of several monomials. Remember that a monomial is an algebraic expression as ax

Factoring 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

2.3. 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

Polynomials and Vieta s Formulas

Polynomials and Vieta s Formulas Misha Lavrov ARML Practice 2/9/2014 Review problems 1 If a 0 = 0 and a n = 3a n 1 + 2, find a 100. 2 If b 0 = 0 and b n = n 2 b n 1, find b 100. Review problems 1 If a

Rational Polynomial Functions

Rational Polynomial Functions Rational Polynomial Functions and Their Domains Today we discuss rational polynomial functions. A function f(x) is a rational polynomial function if it is the quotient of

Dividing Polynomials

4.3 Dividing Polynomials Essential Question Essential Question How can you use the factors of a cubic polynomial to solve a division problem involving the polynomial? Dividing Polynomials Work with a partner.

West Windsor-Plainsboro Regional School District Algebra I Part 2 Grades 9-12

West Windsor-Plainsboro Regional School District Algebra I Part 2 Grades 9-12 Unit 1: Polynomials and Factoring Course & Grade Level: Algebra I Part 2, 9 12 This unit involves knowledge and skills relative

Algebra. Indiana Standards 1 ST 6 WEEKS

Chapter 1 Lessons Indiana Standards - 1-1 Variables and Expressions - 1-2 Order of Operations and Evaluating Expressions - 1-3 Real Numbers and the Number Line - 1-4 Properties of Real Numbers - 1-5 Adding

CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation

CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of

Understanding Basic Calculus

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

476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic

Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

Integrals 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

Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two