4.4 The Rational Root Theorem

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4.4 Rational Root Theorem.

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Presentation transcript:

4.4 The Rational Root Theorem Objectives: Identify all possible rational roots of a polynomial equation by using the Rational Root Theorem. Determine the number of positive and negative real roots a polynomial function has.

Rational Root Theorem: -helps find possible roots (rational) -polynomial a0x^n + a1x^n-1 + …+an-1x + an = 0 q is factors of a0 p is factors of an p/q could be a root. Rational Root Theorem: Ex. 1) List the possible rational roots of 3x³ - 13x² + 2x + 8 = 0. Then find the rational roots. Find the roots of x³ + 6x² - 13x - 6 = 0. Ex. 2)

Descartes’ Rule of Signs: page 231 Used to determine the possible number of positive real zeros a polynomial has. Suppose P(x) is a polynomial whose terms are arranged in descending powers of the variable. Then the # of positive real zeros of P(x) is the same as the # of changes in sign of the coefficients of the terms or is less than this by an even #. The # of negative real zeros of P(x) is the same as the # of changes in sign of the coefficents of the terms of P(-x), or less than this # by an even #. - Ignore zero coefficients.

Ex. 3) Find the # of possible positive real zeros and the # of possible negative real zeros for P(x)=2x^4 - x³ - 2x² + 5x + 1. Then determine the rational zeros. Ex. 4) A manufacturer produces boxes for a calculator company. The boxes have a volume of 240 cm³. Their height is 6 cm less than their width, while their length is 1 cm less than twice their width. Find the dimensions of the box.