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Factoring Polynomials Completely

So far, a typical polynomial has been a product of two factors, with possibly a common factor removed first. However, it is possible that the factors can still be factored again. A polynomial in a single variable may have as many factors as its degree.We have factored a polynomial completely when all of the factors are prime polynomials.


Example 1

Factoring higher-degree polynomials completely

Factor x4 + x2 - 2 completely.


Two numbers with a product of -2 and a sum of 1 are 2 and -1:

x4 + x2 - 2 = (x2 + 2)(x2 - 1)  
  = (x2 + 2)(x - 1)(x + 1) Difference of two squares

Since x2 + 2, x - 1, and x + 1 are prime, the polynomial is factored completely.

In the next example we factor a sixth-degree polynomial.


Example 2

Factoring completely

Factor 3x6 - 3 completely.


To factor 3x6 - 3, we must first factor out the common factor 3 and then recognize that x6 is a perfect square: x6 = (x3)2:

3x6 - 3 = 3(x6 - 1) Factor out the common factor.
  = 3((x3)2 - 1) Write x6 as a perfect square.
  = 3(x3 - 1)(x3 + 1) Difference of two squares
  = 3(x - 1)(x2 + x + 1)(x + 1)(x2 - x + 1) Difference of two cubes and sum of two cubes

Since x2 + x + 1 and x2 - x + 1 are prime, the polynomial is factored completely.

In Example 2 we recognized x6 - 1 as a difference of two squares. However, x6 - 1 is also a difference of two cubes, and we can factor it using the rule for the difference of two cubes:

x6 - 1 = (x2)3 - 1 = (x2 - 1)(x4 + x2 + 1)

Now we can factor x2 - 1, but it is difficult to see how to factor x4  x2 `` 1. (It is not prime.) Although x6 can be thought of as a perfect square or a perfect cube, in this case thinking of it as a perfect square is better.

In the next example we use substitution to simplify the polynomial before factoring. This fourth-degree polynomial has four factors.


Example 3

Using substitution to simplify

Factor (w2 - 1)2 - 11(w2 - 1) + 24 completely.


Let a = w2 - 1 to simplify the polynomial:

(w2 - 1)2 - 11(w2 - 1) + 24 = a2 - 11a + 24 Replace w2 - 1 by a.
  = (a - 8)(a - 3)  
  = (w2 - 1 - 8)(w2 - 1 - 3) Replace a by w2 - 1.
  = (w2 - 9)(w2 - 4)  
  = (w + 3)(w - 3)(w + 2)(w - 2)  
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