Section 6.5Theorems about Roots of

Polynomial EquationsObj: To solve equations using the Rational

Root Theorem

We have learned many methods of solving polynomial equations. Another method, The Rational Root Theorem, involves analyzing one or more integer coefficients of the polynomial in the equation.

Consider: x3 – 5x2 - 2x + 24 = 0 and the equivalent equation (x +2)(x – 3)(x – 4) = 0

The roots are _____ , _____, and _____. The product of these roots is ________.

Notice: All the roots are factors of the ____________.

In general, if the coefficients (including the constant term) in a polynomial equation are integers, then any integer root of the equation is a factor of the ________term.

A similar pattern applies to rational roots.

Remember: A rational number is one that can be written as the quotient of two integers.

Consider: 24x3- 22x2 – 5x + 6 = 0 and its equivalent equation ( x + )( x - )( x - ) = 0

The roots are _______, _______, and _______.

The numerators,1, 2, 3 are all factors of the constant term, _______.

The denominators 2, 3, 4 are all factors of the leading coefficient, _______.

Rational Root TheoremIf simplest form and is a rational root of the polynomial equation

+ + …+ x + = 0with interger coefficients, then p must be a factor of and q must be a factor of .

Which means that play a key role in identifying the rational roots of a polynomial equation.

Also, the real number zeros of a polynomial function are either rational or irrational.

Finding Rational Roots

1. Find the rational roots of x3 + x2 – 3x – 3 = 0

a) List the possible rational roots

b) Test each possible rational root---look for a remainder of 0.

2) Find the possible rational roots of x3 – 4x2 - 2x + 8 = 0

Use the Rational Root Theorem to find all the roots. 1. List the possibilities

2. Test until you find one3. Write quotient4. Factor to find the

remaining roots

1. Find the roots of x3 -2x2 – 5x + 10 = 0

2. Find the roots of 3x3 + x2 –x + 1 = 0

x4 + x3 + x2 - 9x – 10 = 03. Find the roots of:

Closure: If a polynomial equation has integer coefficients, how can you find any rational roots the equation might have?

Apply the Rational Root Theorem and test each possible root.

Assignment: Pg 333 # 1-6 (possibilities only), 7 -12.