Download Intermediate Algebra Section 5.3 – Dividing Polynomials

Intermediate Algebra
Section 5.3 – Dividing Polynomials
When dividing a polynomial by a monomial, we can use the property
of addition of fractions with common denominators.
Dividing a Polynomial by a Monomial
Divide each term in the polynomial by the monomial.
a+b a b
= + , where c ≠ 0 .
c
c c
Example:
Divide the following.
a)
−28 x 4 y
12 x 2 y 3
b)
6x − 3x by 3x
4
3
2
Section 5.3 – Dividing Polynomials
c)
page 2
4x 3 y + 12x 2 y 2 − 4xy 3
4 xy
To divide a polynomial by a polynomial other than a monomial, we
use a method called long division. This method is similar to long
division of real numbers. When using long division for polynomials,
the polynomials must be written in descending order.
Example:
a)
Divide the following.
34 9 7
b) Divide 5x − 5x + 2x + 20 by x + 4 .
2
3
Section 5.3 – Dividing Polynomials
c)
(18x
3
+ x 2 − 90x − 5)÷ (9x 2 − 45)
Definition
If f and g are two functions,
f
is the function defined by
The quotient
g
f ( x)
 f 
, g ( x) ≠ 0 .
 ( x) =
g ( x)
g
Given the functions f ( x ) = x 2 − x − 12 and g ( x ) = x − 4 , find
f 
f 
  ( x ) and   ( 2 )
g
g
page 3
Section 5.3 – Dividing Polynomials
page 4
The Remainder Theorem
Let f be a polynomial function. If f ( x ) is divided by x − c , then
the remainder is f ( c ) .
Example:
Use the Remainder Theorem to find the remainder
when f ( x ) = x 2 + 4 x − 5 is divided by x + 2 .
Section 5.3 – Dividing Polynomials
page 5
The factor Theorem
Let f be a polynomial function. Then x − c is a factor of f ( x ) if
and only if f ( c ) = 0 .
Example:
Use the Factor Theorem to determine whether x + 2 is a
factor of f ( x ) = 3 x 2 + x − 2 . If x + 2 is a factor, then
write f ( x ) in factored form.