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There are two fundamental rules that we follow when factoring:
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:
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.
. This expression is a perfect square trinomial. The and 9 are perfect squares.
The terms that when squared produce and 9 are and 3, respectively.
The middle term is divisible by 2, and . The is the product of and 3, which are the terms that produce the perfect squares.
. This expression is a perfect square trinomial. The and are both perfect squares. The terms that when squared produce and are and , respectively.
The middle term is divisible by . In fact, . Thus,
. This expression is not a perfect square trinomial. Although the middle term is divisible by , the 5 and are not the terms that when squared produce the first and last terms. (This expression would be a perfect square trinomial if the middle term were .)
. This expression is not a perfect square trinomial since the last term is not a perfect square (since any quantity squared is always positive or zero and never negative).
Thus, cannot be factored using this method.
Factor, if possible, the following trinomials.
For the following problems, factor the binomials.
For the following problems, factor, if possible, the trinomials.
( [link] ) Factor by grouping.
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