Perfect Square Binomial: Definition, Lesson & Quiz
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.
We also recommend watching What is the Binomial Theorem? and Multiplying Binomials Using FOIL and the Area Method
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.
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.
In the first step, the first and third term of the trinomial are rewritten as perfect squares. In the next step, the middle term is rewritten as the addition of two identical terms. You can divide the middle term by two to get the two identical terms. These two terms are identified as the multiplication of the perfect squares. Your two perfect squares are 2x and 2y. When you multiply them together, you get 4xy. Because the trinomial breaks down like this, you know that this is a perfect square binomial and your factored form is the square of a binomial where the first and last terms are your perfect squares from the trinomial. Remember, you keep the sign of the middle term in your factored form. If the middle is negative, then your factored form will have a minus.
Try your hand at one more example. See if you can solve it before reading through the steps.
In this example, the middle term is negative, but it can still be broken down into two identical terms, which happen to be the perfect squares multiplied by each other. Do you see how the factored form retains the negative sign?
There is definitely a pattern to perfect square binomials. Because mathematicians like patterns and putting them into formulas, there is a formula for perfect square binomials. It is this:
You can use this formula to multiply out the binomial square or to factor trinomials that are perfect square binomials.
This formula can also be written for a negative middle term like this:
A perfect square binomial is a special case of a trinomial. Not all trinomials are perfect square binomials. Only if the trinomial meets certain criteria can it have such a special label. The criteria include having perfect squares for the first and third term, and the middle term, when divided by 2, must equal the perfect squares multiplied together. Then and only then is the trinomial a perfect square binomial.
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