Download Section 2.4 - Shelton State

Section 2.4
Dividing Polynomials; The Factor
and Remainder Theorems
Overview
• In a previous math experience, we divided
polynomials using long division…
x  3 x  5x  17 x  18
3
2
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
x  5x  3
x 1
4
2
x  7 x  18
x2
2
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) = 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)?
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.
3
2
• Solve the equation x  2 x  5 x  6  0
3
2
given that 2 is a zero of f ( x)  x  2 x  5 x  6