Warm-ups Week 8 10/8/12 Find the zeros of f(x) = x3 + 2x2 – 13x + 10 algebraically (without a graphing calculator). What if I told you that one of the zeros is 2. Can you find the others without using your graphing calculator?

Lesson 2.3: Real Zeros of Polynomial Functions There are two methods for dividing polynomials: long division and synthetic division. Synthetic division is quicker, but long division will work for for divisors of any form.

Long Division Divide 2x4 + 4x3 – 5x2 + 2x – 3 by x2 + 2x – 3.

Synthetic Division Divide x4 – 10x2 + 2x + 3 by x – 3.

The Remainder and Factor Theorems Remainder Theorem: If a polynomial f(x) is divided by (x – k), then the remainder is r = f(k). Factor Theorem: A polynomial f(x) has a factor (x – k) if and only if f(k) = 0.

Example 1 Use the Remainder Theorem to find each remainder and state whether the binomial is a factor, and thus determines a root. Is x - 1 a factor of 2x3 – 3x2 + x ?

Example 2 Divide 3x4 + 4x3 – 12 by x + 2.

Example 3 Divide –x6 + 4x4 + 3x2 + 2x by x + 2.

Example 4 -3 is a root for x3 + 5x2 – 2x – 24. Find the others.

The Rational Root Theorem Sometimes it can be difficult to identify roots of a polynomial. We can use the Rational Root Theorem to give us direction in finding possible roots. If you list the factors of the leading coefficient and the factors of the constant of a polynomial, the possible roots are the factors of the constant divided by the factors of the leading coefficient.

Examples Find the possible rational roots. Then determine the rational roots. x3 + x2 – 5x + 3 6x3 + 11x2 - 3x – 2 x3 + 8x2 + 16x + 5