# Quadratics: Factoring Special Products

When we encounter a polynomial, the first action we take is to simplify that polynomial by factoring. Trinomials, in particular, can be factored into binomials. We have written out the process for factoring a trinomial into binomials, but there is a set of special trinomials (special products) that have memorizable patterns.

If you can recognize special products (in both trinomial and binomial form) you will be able to both factor and develop them quickly and efficiently (which is an especially good skill when preparing for a standardized test, many of which expect students to quickly recognize special products.

There are three types of special products: squares of sums, squares of differences, and the difference of two squares.

Square of Sums | Square of Differences | Difference of Two Squares |

$(x+3)^2=x^2+6x+9$ $(2x+5)^2=4x^2+20x+25$
| $(x-3)^2=x^2-6x+9$ $(2x-5)^2=4x^2-20x+25$ | $(x+3)(x-3)=x^2-9$ $(2x+5)(2x-5)=4x^2-25$ |

The first and last terms of these special product trinomials are the squares of the first and last terms in the binomials. The middle term is $2ac$ (which is just double the last term in the binomial if a=1. Notice that in sum of squares, all signs in the trinomial are positive. | The first and last terms of these special product trinomials are the squares of the first and last terms in the binomials. The middle term is $2ac$ (which is just double the last term in the binomial if a=1. Notice that the only change from the sum of squares is that the middle term is negative. | This special product stands out because the product of two binomials is not a trinomial but a binomial (a difference of two squares). In this case, because one binomial is an addition problem and one is a subtraction problem, when you develop or FOIL these binomials, the middle terms cancel each other out. Note, these special products will always have a product that is a subtraction problem. |

It's worth practicing factoring and developing these special products, to enhance your ability to recognize these special products quickly. If you can recognize, factor, and develop these instantly, you'll do more complex problems much more quickly!

#### Practice Problems:

## Quadratics: Factoring Special Products Practice

Develop (multiply out) the following special products (try to use the patterns rather than foiling -- if you have to foil, pay attention to the patterns!):

- $(x+2)(x+2)$
- $(x-5)(x-5)$
- $(x-3)(x-3)$
- $(x-6)(x-6)$
- $(c-3)^2$
- $(x+12)^2$
- $(x+1)(x-1)$
- $(2x-5)(2x+5)$
- $(3x+4)(3x-4)$
- $(x+y)(x-y)$
- $(x^2+y^2)(x^2-y^2)$
- $(x^2+y^2)^2$

Factor the following trinomials. Do you recognize any special product patterns?

- $x^2+4x+4$
- $x^2-6x+9$
- $x^2-10x+25$
- $x^2-49$
- $4x^2-144$
- $9x^2-16$
- $x^2+20x+100$
- $x^2-16x+64$
- $2x^2-4x+2$
- $3x^2-24x+48$
- $2x^2-12x+18$
- $9x^2-25$