In this explainer, we will learn how to factor a perfect square trinomial expression.
When we are working with algebra, we find that there are many different types of polynomials, that is, expressions that have a number of terms added or subtracted. Let us begin by recapping the definitions of two specific types of polynomials.
Definition: Binomial and Trinomial
A binomial expression has two terms added or subtracted.
A trinomial expression has three terms added or subtracted.
An example of a binomial expression could be .
An example of a trinomial expression could be .
If we have a binomial expression that is squared, we can change it into a trinomial by expanding the parentheses. For example, the binomial expression can be expanded as follows. We can start by writing this as a product of two sets of parentheses:
Next, we can multiply the parentheses using a method such as FOIL or grid. This gives us
A trinomial that is formed from a squared binomial is called a perfect, or perfect square, trinomial. We can define it mathematically as shown below.
Definition: Perfect Square Trinomial
A perfect square trinomial is one that factors into two identical binomial factors. It is written in the form
A perfect square trinomial can be factored as
Therefore, depending on the sign of the middle term, , in our trinomial, we have or
A trinomial is a perfect square trinomial if the first and last terms are positive, perfect squares and the middle term is twice the product of their square roots.
Let us now look at some examples of how we can use the properties of perfect square trinomials to identify and factor them.
Example 1: Recognizing the Conditions for a Perfect Square Trinomial
For which values of is a perfect square?
Answer
Recall that for a trinomial to be a perfect square, it must be in the form .
Therefore, we compare and
Equating the first terms of the two expressions, we have . By taking the positive square roots, we have
Similarly, equating the last terms, we have . Hence, by taking the positive square root, we have
To work out the value of , we equate the middle terms and substitute our values for and , giving us
Since needs to be true for all values of , we have
Therefore, for the trinomial to be a perfect square,
Let us further consider the form that a perfect square trinomial must take, this time by comparing different expressions.
Example 2: Identifying a Perfect Trinomial
Which of the following is a perfect square?
Answer
Since all of the given options are either binomials or trinomials and binomials cannot be perfect squares, we are effectively being asked to identify which of the options is a perfect square trinomial.
Recall that a trinomial is a perfect square trinomial if the first and last terms are positive, perfect squares and the middle term is twice the product of their square roots. It can be written in the form .
Options A and C are binomials; therefore, neither is a perfect square trinomial.
In option B, the last term is not a positive square value; therefore, this is not a perfect square trinomial.
Option D has the first and last terms as positive, perfect squares. The square root of the first term is , and the square root of the last term is 9. However, since the middle term is not equal to twice the product of and 9, it is not a perfect square trinomial.
Option E has the first and last terms as positive, perfect squares. The square root of the first term is , and the square root of the last term is 9. Therefore, we can write as , which is in the form , a perfect square.
If we wanted to write in the factored form , it would be .
So, the answer is option E.
The simplest perfect square trinomials to factor are ones where the first term is a squared variable with a coefficient of 1, such as or . Here is an example of this type.
Example 3: Factoring a Monic Perfect Square Trinomial
Factorize fully .
Answer
Before attempting to factor the expression, let us first consider whether it is in a form that we can easily factor. We note that, here, the first term has a coefficient of 1 and the last term has a coefficient of 25, which are both perfect squares. This suggests that the expression may be in the form of a perfect square trinomial.
Recall that for a trinomial to be a perfect square trinomial, it must be in the form ; it can then be factored as .
Here, we have a negative coefficient of the middle term, so the perfect square trinomial will be in the form . This expression will then factor as .
Therefore, we must compare and
Equating the first terms of the two expressions, we have , and taking the positive square roots gives
Similarly, equating the last terms, we have . Hence, by taking the positive square roots, we get
As a check, we can verify that the middle terms are equal. Substituting for and in the middle term of the general expression, we have as expected. Finally, we write in its factored form , which is .
When given a trinomial to factor, it is important that the terms are in the correct order for comparison with the general form . The three terms should be ordered in descending powers according to the exponents of one of the variables. For instance, in the previous example, the terms were ordered in descending powers of .
Note, however, that using the above convention, we could have ordered the terms in descending powers of by writing them in reverse order as . This would then have resulted in the alternative, equivalent answer . Perfect square trinomials of the form will always have two possible factored forms, depending on the original ordering of the terms.
Let us now consider an example involving a single variable, where the coefficient of the first term is greater than 1.
Example 4: Factoring a Nonmonic Perfect Square Trinomial
Factorize fully .
Answer
Before trying to factor this expression, we need to check whether it is in a recognizable form for factoring. The first term has a coefficient of 81, and the last term has a coefficient of 1. Since both numbers are perfect squares, this suggests that the expression may be in the form of a perfect square trinomial.
Recall that for a trinomial to be a perfect square, it must be in the form ; it can then be factored as .
Here, we have a positive coefficient of the middle term, so the perfect square trinomial will be in the form . This expression will then factor as .
We must compare and
Equating the first terms of the two expressions, we have , and taking the positive square roots gives
Similarly, equating the last terms, we have . Hence, by taking the positive square roots, we get
As a check, we can verify that the middle terms are equal. Substituting for and in the middle term of the general expression, we have as expected. Finally, we write the original expression in its factored form , which is .
In some cases, we will need to take out a common factor from a trinomial expression before we get a perfect square trinomial to factorize. In general, it is useful to look for the highest common factor of the three terms in a given expression. If the highest common factor is greater than 1, we take it out and then apply our usual method to the expression that remains. The next example demonstrates how to do this.
Example 5: Factoring a Trinomial That Results in a Perfect Square Trinomial
Factorize fully .
Answer
Here, the given expression is a trinomial, and we first note that the three terms have a highest common factor of 3, which we can take out to get
Having done this, a sensible strategy to try is to reorder the terms inside the brackets in descending powers of one of the variables. This will help us to see whether it is in a form that we can easily factor. Choosing descending powers of , we have
Observe now that the reordered expression in the brackets has a first term with a coefficient of 1 and a last term with a coefficient of 16, which are both perfect squares. This suggests that the expression is in the form of a perfect square trinomial.
Recall that for a trinomial to be a perfect square, it must be in the form ; it can then be factored as .
In this case, the expression in the brackets has a positive coefficient of the middle term, so this perfect square trinomial will be of the form , which will then factor as .
Therefore, we must compare and
Equating the first terms of the two expressions, we have , and taking the positive square roots gives
Similarly, equating the last terms, we have . Hence, by taking the positive square roots, we get
As a check, we can verify that the middle terms are equal. Substituting for and in the middle term of the general expression, we have which is correct. Thus, we can write the expression in the brackets in its factored form , which is .
Finally, we need to multiply this result by 3, the common factor that we took out of the original expression to get a perfect square trinomial.
So, if we factor the expression fully, we get .
Our final example shows how the properties of perfect square trinomials can be applied to make a numerical calculation easier.
Example 6: Evaluating a Numerical Expression Using Perfect Square Trinomials
By factorizing or otherwise, evaluate .
Answer
Before attempting to factor this expression, let us first consider whether it is in a form that we can easily factor. Note that the first and the last terms are both perfect squares. This suggests that we may be able to rewrite the expression in the form of a perfect square trinomial.
Recall that for a trinomial to be a perfect square, it must be in the form ; it can then be factored as .
In this case, we have a negative coefficient of the middle term, so the perfect square trinomial will be in the form . This expression will then factor as .
With and , we can rewrite the original expression in the perfect trinomial form, , giving us
Putting the right-hand side into the form gives us
Simplifying and evaluating the right-hand side, we have
Therefore, .
Let us finish by recapping some key concepts from this explainer.
Key Points
- A trinomial is a perfect square trinomial if the first and last terms are positive, perfect squares and the middle term is twice the product of their square roots.
- A perfect square trinomial is described mathematically as and can be factored into either or , depending on the sign of the middle term, , in the trinomial.
- In order to compare a given trinomial with the general form , the three terms should be ordered in descending powers according to the exponents of one of the variables.
- For some trinomials, taking out the highest common factor of the three terms results in a perfect square trinomial that can then be factorized.
- In some cases, we can simplify numerical calculations by rewriting them in perfect square trinomial form and factorizing.
