Factoring a Polynomial Completely
Polynomials that cannot be factored are called prime polynomials. Because binomials
such as x 5, a 6, and 3x 1 cannot be factored, they are prime polynomials.
A polynomial is factored completely when it is written as a product of prime
polynomials. To factor completely, always factor out the GCF (or its opposite) first.
Then continue to factor until all of the factors are prime.
Example 1
Factoring completely
Factor each polynomial completely.
a) 5x2 - 20
b) 3a3 - 30a2 + 75a
c) -2b4 + 16b
Solution
| a) 5x2 - 20 |
= 5(x2 - 4) |
Greatest common factor |
| |
= 5(x - 2)(x + 2) |
Difference of two squares |
| b) 3a3 - 30a2 + 75a |
= 3a(a2 -10a + 25) |
Greatest common factor |
| |
= 3a(a - 5)2 |
Perfect square trinomial |
| c) -2b4 + 16b |
= -2b(b3 - 8) |
Factor out 2b to make
the next step easier. |
| |
= -2b(b - 2)(b2 + 2b + 4) |
Difference of two cubes |
