1. 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.

2. Rational Root Theorem:
-helps find possible roots (rational)
-polynomial a0x^n + a1x^n-1 + …+an-1x + an = 0
q is factors of a 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)

3. 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.

4. Ex. 3)
Find the # of possible positive real zeros and the # of possible negative real zeros for
P(x)=2x^4 - x³ - 2x² + 5x 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.