Section 2.4 Dividing Polynomials; The Factor and Remainder Theorems

Overview In a previous math experience, we divided polynomials using long division… Let’s label the parts of this division problem.

Synthetic Division Can be used to divide a polynomial P(x) by a linear divisor x – r

Examples

Important Stuff Don’t forget to put in zeros for the missing terms in your dividend. The “answers” are the coefficients of your quotient, except for the last number, which is your remainder. The degree of the quotient is always one degree less than the degree of the dividend.

A Little Bit of Function Review If f(x) = x 3 – 5x x – 18, what is f(-3)? If g(x) = x 4 – 5x 2 – 3, what is g(1)? If h(x) = x 2 – 7x – 18, what is h(-2)?

The relationship between synthetic division and evaluating a polynomial function The Remainder Theorem: if the polynomial f(x) is divided by x – r, then the remainder is f(r). English Translation: when you divide using synthetic division, your remainder is the same as what you would get if you evaluated the function using the number in the box.

The significance of a zero remainder We say that a number x is a factor of another number y when dividing y by x yields a remainder of 0. The same idea applies to dividing polynomials: If dividing f(x) by x – r gives a 0 remainder, then by the Remainder Theorem f(r) = 0.

The Factor Theorem This makes x – r a factor of f(x). Important definition: a number r is a zero (or root) of a polynomial f(x) when f(r) = 0. If we were to graph f(x), the point (r,0) would be an x-intercept.

Pop Quiz Name the three ways to solve a quadratic equation. Solve the equation given that 2 is a zero of