- The Difference of Two Squares
- Fundamental Rules of Factoring
- Perfect Square Trinomials

Inside Collection (Textbook): Basic Mathematics Review

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Factoring is an essential skill for success in algebra and higher level mathematics courses. Therefore, we have taken great care in developing the student's understanding of the factorization process. The technique is consistently illustrated by displaying an empty set of parentheses and describing the thought process used to discover the terms that are to be placed inside the parentheses. The factoring scheme for special products is presented with both verbal and symbolic descriptions, since not all students can interpret symbolic descriptions alone. Two techniques, the standard "trial and error" method, and the "collect and discard" method (a method similar to the "ac" method), are presented for factoring trinomials with leading coefficients different from 1. Objectives of this module: know the fundamental rules of factoring, be able to factor the difference of two squares and perfect square trinomials.

- The Difference of Two Squares
- Fundamental Rules of Factoring
- Perfect Square Trinomials

Recall that when we multiplied together the two binomials

Notice that the terms *difference* of the two squares.

Since we know that

The factorization form says that we can factor

When using real numbers (as we are), there is no factored form for the sum of two squares. That is, using real numbers,

Factor

We can check our factorization simply by multiplying.

We can check our factorization by multiplying.

Now we see that

Be careful not to drop the factor 3.

If possible, factor the following binomials completely.

There are two fundamental rules that we follow when factoring:

- Factor out all common monomials first.
- Factor completely.

Factor each binomial completely.

Now we can see a difference of two squares, whereas in the original polynomial we could not. We’ll complete our factorization by factoring the difference of two squares.

Finally, the factorization is complete.

These types of products appear from time to time, so be aware that you may have to factor more than once.

Factor each binomial completely.

Recall the process of squaring a binomial.

Our Method Is | We Notice |

Square the first term. | The first term of the product should be a perfect square. |

Take the product of the two terms and double it. | The middle term of the product should be divisible by 2 (since it’s multiplied by 2). |

Square the last term. | The last term of the product should be a perfect square. |

Perfect square trinomials *always* factor as the square of a binomial.

To recognize a perfect square trinomial, look for the following features:

- The first and last terms are perfect squares.
- The middle term is divisible by 2, and if we divide the middle term in half (the opposite of doubling it), we will get the product of the terms that when squared produce the first and last terms.

In other words, factoring a perfect square trinomial amounts to finding the terms that, when squared, produce the first and last terms of the trinomial, and substituting into one of the formula

Factor each perfect square trinomial.

The terms that when squared produce

The middle term is divisible by 2, and

The middle term

*not* a perfect square trinomial. Although the middle term is divisible by

*not* a perfect square trinomial since the last term

Thus,

Factor, if possible, the following trinomials.

not possible

For the following problems, factor the binomials.

For the following problems, factor, if possible, the trinomials.

not factorable

*((Reference))* Factor

*((Reference))* Factor

- « Previous module in collection Factoring by Grouping
- Collection home: Basic Mathematics Review
- Next module in collection » Factoring Trinomials with Leading Coefficient 1

Comments:"Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"