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# CollegeAlgebra MATH0012

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This 3 page Class Notes was uploaded by Aileen Davis on Tuesday October 20, 2015. The Class Notes belongs to MATH0012 at Sierra College taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/225383/math0012-sierra-college in Mathematics (M) at Sierra College.

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Date Created: 10/20/15

MATH 12 NOTES SECTIONS 33 AND 34 Ratti 33 We previoust discussed long division of polynomials and synthetic division Let s look at two more synthetic division examples since synthetic division will be the more useful procedure to help us nd zeros of polynomials 1 Divide 3X32X2X 15 by X 3 2 Divide 3X46X23X30 by X2 There are a few important results in the remainder of section 33 The Remainder Theorem lfa polynomial FX is divided by X a then the remainder R is given by Fa This can provide a quick way to evaluate polynomials and to give us points on the graph of y FX The Factor Theorem A polynomial FX has X a as a factor ifand only if Fa 0 As a result of the Factor Theorem the following problems are equivalent 0 Factoring FX 0 Finding the zeros of the function FX de ned by a polynomial expression 0 Solving or finding the roots of the polynomial equation FX 0 The Rational Zeros Test lf FX anx ani1X 1 a1Xa0 is a polynomial function with integer coef cients and Bis a rational number in lowest terms that q is a zero of FX then p is a factor of a0 and q is a factor of a This result will give us a list of possible rational zeros for our polynomial Then we will test to find out which if any are zeros In doing so we will also be able to factor our polynomial Let s look at some examples Examples provided in class 34 lfwe can determine a list of possible zeros and we know how to determine when we have a zero of a polynomial function how do we know how many zeros to look for Are they positive or negative Can we impose any boundaries on these zeros to help us re ne our search Remember that a polynomial function of degree n has at most n real zeros Descartes s Rule of Signs will help us determine how many positive and how many negative zeros there are The number of positive zeros of FX is either equal to the number of sign changes of FXor less than that number by an even integer The number of negative zeros of FX is either equal to the number of sign changes of F X or less than that number by an even integer When using Descartes s Rule of Signs a zero of multiplicity m should be counted at m zeros Con39ugate Pairs Theorem Imaginary zeros of FX occur in conjugate pairs Upper and Lower Bounds on Zeros FX is a polynomial with real coef cients and a positive leading coef cient FX is synthetically divided by X k If k gt 0 and each number in the last row is either positive or zero then k is an upper bound on the zeros of FX If k lt0 and the numbers in the last row alternative in sign then k is a lower bound on the zeros of FX Just a few more theorems and we ll be ready to do some problems FUNDAMENTAL THEOREM OF ALGEBRA Every polynomial with complex coef cients has at least one complex zero FACTORIZATION THEOREM FOR POLYNOMIALS lf PX is a complex polynomial of degree n 21 it can be factored into n not necessarily distinct linear factors ofthe form PX ax r1x r2x r3x rn where a r1 r2 r are complex numbers We can now re ne the result from section 32 A polynomial of degree n has exactly n zeros Finally everv polvnomial with real coef cients can be uniquely factored overthe real numbers as a product of linear and irreducible guadratic factors

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