1
6.5

• Factoring Binomials

2
Difference of Two Squares

• Another shortcut for factoring a trinomial is
  when we want to factor the difference of two
  squares.
• a2 b2 (a b)(a b)
• A binomial is the difference of two squares if
• both terms are squares and
• the signs of the terms are different.

9x2 25y2
c4 d4
3
Difference of Two Squares
Example
Factor the polynomial x2 9. The first term is
a square and the last term, 9, can be written as
32. The signs of each term are different, so we
have the difference of two squares Therefore x2
9 (x 3)(x 3). Note You can use FOIL
method to verify that the factorization for the
polynomial is accurate.
4
Sum or Difference of Two Cubes

• There are two additional types of binomials that
  can be factored easily by remembering a formula.
  
• We have not studied these special products
  previously, as they involve cubes of terms,
  rather than just squares.
• a3 b3 (a b)(a2 ab b2)
• a3 b3 (a b)(a2 ab b2)

5
Sum or Difference of Two Cubes
Example

• Factor x3 1.
• Since this polynomial can be written as x3 13,
• x3 1 (x 1)(x2 x 1).
• Factor y3 64.
• Since this polynomial can be written as y3 43,
• y3 64 (y 4)(y2 4y 16).

6
Sum or Difference of Two Cubes
Example

• Factor 8t3 s6.
• Since this polynomial can be written as (2t)3
  (s2)3,
• 8t3 s6 (2t s2)((2t)2 (2t)(s2) (s2)2)
• (2t s2)(4t2 2s2t s4).
• Factor x3y6 27z3.
• Since this polynomial can be written as (xy2)3
  (3z)3,
• x3y6 27z3 (xy2 3z)((xy2)2 (3z)(xy2)
  (3z)2)
• (xy2 3z)(x2y4 3xy2z 9z2).