Factoring Perfect Square Trinomials Practice Problems

Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

A perfect square trinomial is a special form of trinomial that has a unique method for factoring it. We'll go over that method here and then cement it in your mind by working through a couple practice problems.

What is a Perfect Square Trinomial?

In an algebra course, you'll spend a lot of time working with polynomials. We rank these polynomials according to the number of terms in them. For example, a trinomial is a polynomial that has three terms.


trinomial terms


What you'll find you spend most of your time doing with trinomials is factoring them in order to solve for the unknowns. In this lesson, we're going to learn how to factor a special kind of trinomial called a perfect square trinomial. Perfect square trinomials come in the following two forms:


perfect square trinomial formula


To the left of the equals sign is the trinomial you'll be given, and to the right is what the factored form of it looks like. The terms that make up our perfect square trinomial can be broken up into the components a and b. These two components can consist of constants, unknowns, or a combination of the two.

How to Identify a Perfect Square Trinomial

Now you know in theory what a perfect square trinomial is, but how do you tell if the specific trinomial you're working with is one? Remember, you won't have what's on the right side of the equals sign in the equations above. That's what you're trying to find.

In order to see if a trinomial you're working with is a perfect square trinomial you need to check for the following three conditions:

Condition 1: The first and third terms must be positive.

Condition 2: The first and third terms must be perfect squares.

Condition 3: If the second term is positive it's equal to 2 times the square roots of the first and third term. If the second term is negative it's equal to -2 times the square roots of the first and third term.

You have a perfect square trinomial if and only if all three conditions are met.

Practice Problems

The best way to get used to working with perfect square trinomials is by working through practice problems. Let's factor a couple trinomials together.

Problem One

We'll start by going through our checklist to see if the following trinomial is a perfect square trinomial.


problem 1 trinomial


Condition 1: Are the first and third terms both positive? Our first term is 4x^2 and our third is 25. Both are positive so this condition is passed.

Condition 2: Are the first and third terms both perfect squares? To determine this, we take their square roots. If they are perfect squares, constants should come out as whole numbers, and the unknowns' exponents should end up as whole numbers as well.


problem 1 part 1


Both of our terms pass this test as well. If this trinomial turns out to be a perfect square trinomial, 2x will be our a and 5 will be our b from the formulas for factoring a perfect square trinomial.

Condition 3: Is the second term equal to two times the square roots of the first and third terms?


problem 1 part 2


Our trinomial has now passed all three conditions. So we can use a formula for factoring a perfect square trinomial on it.

First, we need to check which of the two versions of the formula we are going to use. This depends on whether the second term is positive or negative. Since it's positive (20x) we'll use the following formula:


problem 1 part 3


We've already seen that for this trinomial a = 2x and b = 5. Knowing this, we get the following answer for our fully factored trinomial.


problem 1 splution


Problem Two

For our second practice problem, let's look at something a little more complicated.


problem 2 trinomial


Immediately we can see this fails our first condition as both the first and third terms are negative. However, you shouldn't give up so fast. We can try factoring out a negative one from the equation.


problem 2 part 1


Now both the first and third terms inside the parentheses are positive. We can continue checking if what's inside the parentheses is a perfect square trinomial.

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