FACTORING POLYNOMIALS
(Remainder & Factor Theorems)
Ex 
a)
Divide f(x) = –x4 + 3x3 – 10x – 6 by (x + 2). What is the remainder?
b)
Evaluate f(–2). What do you notice
The Remainder Theorem:
When a polynomial, f(x), is divided by (x – a),
the remainder is equal to f(a).
The Factor Theorem is a special case of the Remainder Theorem and it is used to
factor polynomials (of degree 3 or greater)!!
The Factor Theorem:
(x – a) is a factor of f(x), if and only if f(a) = 0
NOTE: iff means (x – a) is a factor of f(x) if f(a) = 0 and if f(a) = 0 then (x – a) is a factor of f(x)
Ex 
Determine whether the following are factors of f(x) = x3 – 6x2 + 3x + 10.
a)
(x – 1)
c)
(2x – 1)
b)
(x + 1)
Ex 
Fully factor each of the following polynomials using the factor theorem:
a)
f(x) = x3 – 5x2 – 2x + 24
b)
f(x) = –x4 + 4x3 + 3x2 – 10x – 8
Try factors of 24!!
SUMMARY

To factor a polynomial, f(x), of degree 3 or greater:
 use the Factor Theorem to determine a factor of f(x)
 divide f(x) by (x – a)
 factor the quotient, if possible

If a polynomial has a degree greater than 3, it may be necessary to use
the Factor Theorem more than once.

Not all polynomial functions are factorable.
Ex 
When f(x) = 2x3 – kx2 + 5x – 2 is divided by (x + 1) the remainder is –12.
Determine the value of k.
Ex 
When f(x) = 2x3 – kx2 + mx – 2 is divided by (x – 1) the remainder is –12.
Determine the value of k and m if (x – 2) is a factor of f(x).
Homework:
p.177  4ace, 5ab, 6acf, 7ad, 8ad,
9, 10, 13