Division and Factoring
To factor a polynomial means to write it as a product of two or more simpler polynomials.
If we divide two polynomials and get 0 remainder, then we can write
dividend = (divisor)(quotient)
and we have factored the dividend. The dividend factors as the divisor times the
quotient if and only if the remainder is 0. We can use division to help us discover
factors of polynomials. To use this idea, however, we must know a factor or a
possible factor to use as the divisor.
Example 1
Using synthetic division to determine factors
Is x  1 a factor of 6x^{3}  5x^{2}  4x + 3?
Solution
We can use synthetic division to divide 6x^{3}  5x^{2}  4x
+ 3 by x  1:
1 
6 
5 
4 
3 

↓ 
6 
1 
3 

6 
1 
3 
0 
Because the remainder is 0, x  1 is a factor, and
6x^{3}  5x^{2}  4x + 3 = (x  1)(6x^{2} + x
 3).
Example 2
Using division to determine factors
Is a  b a factor of a^{3}  b^{3}?
Solution
Divide a^{3}  b^{3 }by a  b. Insert zeros for the missing a^{2}b and ab^{2}terms.
Because the remainder is 0, a  b is a factor, and
a^{3}  b^{3} = (a  b)(a^{2} + ab + b^{2}).
