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1.5 Factoring polynomials  (Page 3/6)

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Factor a.   2 x 2 + 9 x + 9   b.   6 x 2 + x − 1

a.   ( 2 x + 3 ) ( x + 3 )   b.   ( 3 x −1 ) ( 2 x + 1 )

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Factoring a perfect square trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

a 2 + 2 a b + b 2 = ( a + b ) 2 and a 2 − 2 a b + b 2 = ( a − b ) 2

We can use this equation to factor any perfect square trinomial.

Perfect square trinomials

A perfect square trinomial can be written as the square of a binomial:

a 2 + 2 a b + b 2 = ( a + b ) 2

Given a perfect square trinomial, factor it into the square of a binomial.

  1. Confirm that the first and last term are perfect squares.
  2. Confirm that the middle term is twice the product of   a b .
  3. Write the factored form as   ( a + b ) 2 .

Factoring a perfect square trinomial

Factor   25 x 2 + 20 x + 4.

Notice that   25 x 2   and   4   are perfect squares because   25 x 2 = ( 5 x ) 2   and   4 = 2 2 .   Then check to see if the middle term is twice the product of   5 x   and   2.   The middle term is, indeed, twice the product:   2 ( 5 x ) ( 2 ) = 20 x .   Therefore, the trinomial is a perfect square trinomial and can be written as   ( 5 x + 2 ) 2 .

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Factor   49 x 2 − 14 x + 1.

( 7 x −1 ) 2

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Factoring a difference of squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.

a 2 − b 2 = ( a + b ) ( a − b )

We can use this equation to factor any differences of squares.

Differences of squares

A difference of squares can be rewritten as two factors containing the same terms but opposite signs.

a 2 − b 2 = ( a + b ) ( a − b )

Given a difference of squares, factor it into binomials.

  1. Confirm that the first and last term are perfect squares.
  2. Write the factored form as   ( a + b ) ( a − b ) .

Factoring a difference of squares

Factor   9 x 2 − 25.

Notice that   9 x 2   and   25   are perfect squares because   9 x 2 = ( 3 x ) 2   and   25 = 5 2 .   The polynomial represents a difference of squares and can be rewritten as   ( 3 x + 5 ) ( 3 x − 5 ) .

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Factor   81 y 2 − 100.

( 9 y + 10 ) ( 9 y − 10 )

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Is there a formula to factor the sum of squares?

No. A sum of squares cannot be factored.

Factoring the sum and difference of cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 )

Similarly, the sum of cubes can be factored into a binomial and a trinomial, but with different signs.

a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 )

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: S ame O pposite A lways P ositive. For example, consider the following example.

x 3 − 2 3 = ( x − 2 ) ( x 2 + 2 x + 4 )

The sign of the first 2 is the same as the sign between   x 3 − 2 3 .   The sign of the   2 x   term is opposite the sign between   x 3 − 2 3 .   And the sign of the last term, 4, is always positive .

Sum and difference of cubes

We can factor the sum of two cubes as

a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 )

We can factor the difference of two cubes as

a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 )

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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