Lesson Explainer: Factoring Perfect-Square Trinomials Mathematics

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 5𝑥7𝑦.

An example of a trinomial expression could be 3𝑥2𝑥+5.

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 (2𝑥3) can be expanded as follows. We can start by writing this as a product of two sets of parentheses as follows: (2𝑥3)=(2𝑥3)(2𝑥3).

Next, we can multiply the parentheses, using a method such as FOIL or grid. This gives us (2𝑥3)=(2𝑥3)(2𝑥3)=4𝑥6𝑥6𝑥+9=4𝑥12𝑥+9.

A trinomial that is formed from a squared binomial is called a perfect, or perfect-square, trinomial. We can define it mathematically as below.

Definition: Perfect Trinomial

A perfect trinomial is one that factors into two identical binomial factors. It is written in the form 𝑎±2𝑎𝑏+𝑏.

A perfect trinomial can be factored as (𝑎±𝑏).

Therefore, depending on the sign of the middle term, 2𝑎𝑏, in our trinomial, we have 𝑎+2𝑎𝑏+𝑏=(𝑎+𝑏) or 𝑎2𝑎𝑏+𝑏=(𝑎𝑏).

A trinomial is a perfect 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 trinomials to identify and factor them.

Example 1: Identifying a Perfect Trinomial

Which of the following is a perfect square?

  1. 𝑥18𝑥+81
  2. 𝑥18𝑥81
  3. 𝑥9𝑥+81
  4. 𝑥81
  5. 𝑥+81

Answer

Recall that a trinomial is a perfect 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 𝑎±2𝑎𝑏+𝑏.

Option A has the first and last terms as positive, perfect squares. The square root of the first term is 𝑥, and the square root of the last term is 9. Therefore, we can write 𝑥18𝑥+81 as (𝑥)2(𝑥)(9)+(9), which is in the form 𝑎2𝑎𝑏+𝑏, a perfect square.

If we wanted to write 𝑥18𝑥+81 in the factored form (𝑎𝑏), it would be (𝑥9).

In option B, the last term is not a positive square value; therefore, it is not a perfect-square trinomial.

Option C has the first and last terms as positive, perfect squares. The square root of the first term is 𝑥 and the square root of the last term is 9. However, since the middle term is not equal to twice the product of 𝑥 and 9, it is not a perfect-square trinomial.

Options D and E are binomials; therefore, neither is a perfect-square trinomial.

So, the answer is option A.

Example 2: Finding the Missing Value in a Perfect Trinomial

For which values of 𝑘 is 16𝑥+𝑘𝑥+81 a perfect square?

Answer

Recall that for a trinomial to be a perfect square it must be in the form 𝑎±2𝑎𝑏+𝑏.

Therefore, we compare 16𝑥+𝑘𝑥+81 and 𝑎±2𝑎𝑏+𝑏.

Equating the first terms of the two expressions, we have 𝑎=16𝑥. By taking the positive square roots, we have 𝑎=4𝑥.

Similarly, equating the last terms, we have 𝑏=81. Hence, by taking the positive square root, we have 𝑏=9.

To work out the value of 𝑘, we equate the middle terms and substitute our values for 𝑎 and 𝑏, giving us 𝑘𝑥=±2𝑎𝑏=±2(4𝑥)(9)=±72𝑥.

Since 𝑘𝑥=±72𝑥 needs to be true for all values of 𝑥, we have 𝑘=±72.

Therefore, for the trinomial 16𝑥+𝑘𝑥+81 to be a perfect square, 𝑘=72𝑘=72.or

Example 3: Finding the Missing Value in a Perfect Trinomial

Complete the expression 60𝑥+25 to make a perfect square.

Answer

Recall that for a trinomial to be a perfect trinomial it must be in the form 𝑎±2𝑎𝑏+𝑏.

Here, we have a negative coefficient of the middle term, so the perfect trinomial will be in the form 𝑎2𝑎𝑏+𝑏.

Therefore, we compare 60𝑥+25 and 𝑎2𝑎𝑏+𝑏.

Equating the last terms, we have 𝑏=25.

So, 𝑏=5.

Next, we can equate the middle terms, giving us 2𝑎𝑏=60𝑥2𝑎𝑏=60𝑥.

We can substitute in 𝑏=5 and simplify, giving us 2𝑎𝑏=60𝑥2𝑎(5)=60𝑥10𝑎=60𝑥.

Dividing both sides by 10 gives us 𝑎=6𝑥.

Therefore, 𝑎=36𝑥.

So, the completed expression to make a perfect square is 36𝑥60𝑥+25.

We will now look at an example of how the properties of perfect trinomials can be applied to make a numerical calculation easier.

Example 4: Factoring a Perfect Trinomial

By factoring, or otherwise, evaluate (10.1)4.2×10.1+(2.1).

Answer

We can use the properties of a perfect trinomial to solve this problem. Recall that 𝑎2𝑎𝑏+𝑏=(𝑎𝑏).

With 𝑎=10.1 and 𝑏=2.1, we can write the given expression in the perfect trinomial form, 𝑎2𝑎𝑏+𝑏, giving us (10.1)4.2×10.1+(2.1)=(10.1)2×(10.1)(2.1)+(2.1).

Putting the right-hand side into the form (𝑎𝑏) gives us (10.1)4.2×10.1+(2.1)=(10.12.1).

Simplifying and evaluating the right-hand side, we have (10.1)4.2×10.1+(2.1)=8=64.

Therefore, (10.1)4.2×10.1+(2.1)=64.

We will see in the next example how we can apply these algebraic skills with perfect trinomials to a geometric problem.

Example 5: Factoring a Perfect Trinomial

A square has an area of 81𝑐+54𝑐𝑑+9𝑑. What is the length of its side?

Answer

We can use the properties of a perfect trinomial to solve this problem. Recall that 𝑎+2𝑎𝑏+𝑏=(𝑎+𝑏).

By setting 𝑎=9𝑐 and 𝑏=3𝑑, we can write the given expression in the perfect trinomial form 𝑎+2𝑎𝑏+𝑏, giving us 81𝑐+54𝑐𝑑+9𝑑=9𝑐+2×9𝑐(3𝑑)+(3𝑑).

Putting the right-hand side of the equation into the form (𝑎+𝑏) gives us 81𝑐+54𝑐𝑑+9𝑑=9𝑐+3𝑑.

Therefore, since the area of a square is given by its side length squared, we can say that the length of the side is ||9𝑐+3𝑑||.

Key Points

  • A trinomial is a perfect 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 trinomial is mathematically described as 𝑎±2𝑎𝑏+𝑏 and can be factored into either (𝑎+𝑏) or (𝑎𝑏), depending on the sign of the middle term, 2𝑎𝑏, in the trinomial

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