Dividing Polynomials; The Factor and Remainder Theorems

1
Section 2.4

• Dividing Polynomials The Factor and Remainder
  Theorems

2
Overview

• In a previous math experience, we divided
  polynomials using long division

Lets label the parts of this division problem.
3
Synthetic Division

• Can be used to divide a polynomial P(x) by a
  linear divisor x r

4
Examples
5
Important Stuff

• Dont 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.

6
A Little Bit of Function Review

• If f(x) x3 5x2 17x 18, what is f(-3)?
• If g(x) x4 5x2 3, what is g(1)?
• If h(x) x2 7x 18, what is h(-2)?

7
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.

8
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.

9
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.

10
Pop Quiz

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