SOLVING POLYNOMIAL EQUATIONS


 Rhoda Sullivan
 6 years ago
 Views:
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 firstdegree 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 lowerdegree 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 xintercept 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 fifthdegree 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 fifthdegree 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
More informationJUST 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 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 informationApplication. Outline. 31 Polynomial Functions 32 Finding Rational Zeros of. Polynomial. 33 Approximating Real Zeros of.
Polynomial and Rational Functions Outline 31 Polynomial Functions 32 Finding Rational Zeros of Polynomials 33 Approximating Real Zeros of Polynomials 34 Rational Functions Chapter 3 Group Activity:
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 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 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 informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
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 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 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 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 n1 x n1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role
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 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 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 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 xintercept) of a polynomial is identical to the process of factoring a polynomial.
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 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 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 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 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 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 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 information2.5 ZEROS OF POLYNOMIAL FUNCTIONS. Copyright Cengage Learning. All rights reserved.
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.
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 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 informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + 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 informationPolynomials. 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
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 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 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 informationPolynomial Expressions and Equations
Polynomial Expressions and Equations This is a really closeup picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen
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 informationThe 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
More informationSECTION 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
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 (927) 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 27in. wheel, 44 teeth
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 informationInteger 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
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 informationSection 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)
More informationUnit 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
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 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 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 informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 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 informationFactoring 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
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 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 informationSOLVING POLYNOMIAL EQUATIONS BY RADICALS
SOLVING POLYNOMIAL EQUATIONS BY RADICALS Lee Si Ying 1 and Zhang DeQi 2 1 Raffles Girls School (Secondary), 20 Anderson Road, Singapore 259978 2 Department of Mathematics, National University of Singapore,
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 informationCubic Functions: Global Analysis
Chapter 14 Cubic Functions: Global Analysis The Essential Question, 231 Concavitysign, 232 Slopesign, 234 Extremum, 235 Heightsign, 236 0Concavity Location, 237 0Slope Location, 239 Extremum Location,
More informationcalculating 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
More informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
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 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 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 informationPartial 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
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 informationCM2202: 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
More information7.2 Quadratic Equations
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
More informationUnderstanding 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
More informationWinter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com
Polynomials Alexander Remorov alexanderrem@gmail.com Warmup 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 informationCopyrighted 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
More informationPartial Fractions. (x 1)(x 2 + 1)
Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +
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 informationH/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
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 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 informationSECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
More 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 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 informationSome 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,
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 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 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 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 informationRoots of Polynomials
Roots of Polynomials (Com S 477/577 Notes) YanBin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x
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 information3 Factorisation into irreducibles
3 Factorisation into irreducibles Consider the factorisation of a nonzero, noninvertible integer n as a product of primes: n = p 1 p t. If you insist that primes should be positive then, since n could
More informationThe Notebook Series. The solution of cubic and quartic equations. R.S. Johnson. Professor of Applied Mathematics
The Notebook Series The solution of cubic and quartic equations by R.S. Johnson Professor of Applied Mathematics School of Mathematics & Statistics University of Newcastle upon Tyne R.S.Johnson 006 CONTENTS
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 informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
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 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 informationFACTORING POLYNOMIALS
296 (540) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
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 informationis the degree of the polynomial and is the leading coefficient.
Property: T. HrubikVulanovic email: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 HigherDegree Polynomial Functions... 1 Section 6.1 HigherDegree Polynomial Functions...
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 informationSOLUTIONS FOR PROBLEM SET 2
SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such
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 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 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 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 information4.3 Lagrange Approximation
206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average
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 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 information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on realclosed fields These notes develop the algebraic background needed to understand the model theory of realclosed fields. To understand these notes, a standard graduate course in algebra is
More information15. Symmetric polynomials
15. Symmetric polynomials 15.1 The theorem 15.2 First examples 15.3 A variant: discriminants 1. The theorem Let S n be the group of permutations of {1,, n}, also called the symmetric group on n things.
More information