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Perfect Square Binomial: Definition & Explanation

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught Math at a public charter high school.

Learn what sets perfect square binomials apart from other trinomials in the math world. You will also learn an easy method to identify them along with a simple procedure to factor them.

Definition

A perfect square binomial is a trinomial that when factored gives you the square of a binomial. For example, the trinomial x^2 + 2xy + y^2 is a perfect square binomial because it factors to (x + y)^2. Do you notice how the trinomial in factored form is the square of a binomial? Also, look at the first and last term of the trinomial. Do you notice something interesting about them? Both terms are perfect squares. That is one indication that the trinomial you are dealing with may be a perfect square binomial.

Here are some more examples of special case trinomials that are perfect square binomials.

Examples of trinomials that are perfect square binomials.
perfect square binomials

Do you see a pattern in these trinomials and their respective factored forms? Do you see how these are special trinomials? This is what sets them apart from other trinomials. It actually makes factoring them easier if you know that they are a perfect square binomial before you begin to factor.

An easy way to check whether a trinomial is a perfect square binomial is to look at the first and third term to see if they are perfect squares. If they are, then check the second term by dividing it by 2. The result should be the two perfect squares multiplied by each other. For example, the trinomial x^2 + 2xy + y^2 has perfect squares for the first and third term. The first term is (x)^2 and the third term is (y)^2. Multiply the two squares together and you get xy. When you divide the middle term by 2, you should get xy.

Notice the third trinomial in the list and you will see that the middle term is a negative. The middle term can be either positive or negative. The negative sign determines the sign of the factored form.

How to Factor Perfect Square Binomials

Once you have identified the trinomial as a perfect square binomial, factoring it becomes very easy. Recall the patterns you saw in the examples. What did you notice about the factored forms of the trinomials? They had the squares as the first and second term. So, factoring a perfect square binomial requires you to know your squares table (1x1=1; 2x2=4; 3x3=9; etc.). If you haven't already memorized them, start reviewing them now.

Look at this factoring example, and see if you can follow the steps.

Can you follow the steps?
perfect square binomial factoring

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