Explainer: Factoring Perfect-Square Trinomials

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 5π‘₯4βˆ’7𝑦.

An example of a trinomial expression could be 3π‘₯2βˆ’2π‘₯+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 (2π‘₯βˆ’3)2 can be expanded as follows. We can start by writing this as a product of two sets of parentheses as follows: (2π‘₯βˆ’3)2=(2π‘₯βˆ’3)(2π‘₯βˆ’3).

Next, we can multiply the parentheses, using a method such as FOIL or grid. This gives us (2π‘₯βˆ’3)2=(2π‘₯βˆ’3)(2π‘₯βˆ’3)=4π‘₯2βˆ’6π‘₯βˆ’6π‘₯+9=4π‘₯2βˆ’12π‘₯+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 π‘Ž2Β±2π‘Žπ‘+𝑏2.

A perfect trinomial can be factored as (π‘ŽΒ±π‘)2.

Therefore, depending on the sign of the middle term, 2π‘Žπ‘, in our trinomial, we have π‘Ž2+2π‘Žπ‘+𝑏2=(π‘Ž+𝑏)2 or π‘Ž2βˆ’2π‘Žπ‘+𝑏2=(π‘Žβˆ’π‘)2.

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. π‘₯2βˆ’18π‘₯+81
  2. π‘₯2βˆ’18π‘₯βˆ’81
  3. π‘₯2βˆ’9π‘₯+81
  4. π‘₯2βˆ’81
  5. π‘₯2+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 π‘Ž2Β±2π‘Žπ‘+𝑏2.

Option A has the first and last terms as positive, perfect squares. The square root of the first term is π‘₯, and the square root of the last term is 9. Therefore, we can write π‘₯2βˆ’18π‘₯+81 as (π‘₯)2βˆ’2(π‘₯)(9)+(9)2, which is in the form π‘Ž2βˆ’2π‘Žπ‘+𝑏2, a perfect square.

If we wanted to write π‘₯2βˆ’18π‘₯+81 in the factored form (π‘Žβˆ’π‘)2, it would be (π‘₯βˆ’9)2.

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 π‘₯ and the square root of the last term is 9. However, since the middle term is not equal to twice the product of π‘₯ 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 π‘˜ is 16π‘₯2+π‘˜π‘₯+81 a perfect square?

Answer

Recall that for a trinomial to be a perfect square it must be in the form π‘Ž2Β±2π‘Žπ‘+𝑏2.

Therefore, we compare 16π‘₯2+π‘˜π‘₯+81 and π‘Ž2Β±2π‘Žπ‘+𝑏2.

Equating the first terms of the two expressions, we have π‘Ž2=16π‘₯2. By taking the positive square roots, we have π‘Ž=4π‘₯.

Similarly, equating the last terms, we have 𝑏2=81. Hence, by taking the positive square root, we have 𝑏=9.

To work out the value of π‘˜, we equate the middle terms and substitute our values for π‘Ž and 𝑏, giving us π‘˜π‘₯=Β±2π‘Žπ‘=Β±2(4π‘₯)(9)=Β±72π‘₯.

Since π‘˜π‘₯=Β±72π‘₯ needs to be true for all values of π‘₯, we have π‘˜=Β±72.

Therefore, for the trinomial 16π‘₯2+π‘˜π‘₯+81 to be a perfect square, π‘˜=72orπ‘˜=βˆ’72.

Example 3: Finding the Missing Value in a Perfect Trinomial

Complete the expression βˆ’60π‘₯2+25 to make a perfect square.

Answer

Recall that for a trinomial to be a perfect trinomial it must be in the form π‘Ž2Β±2π‘Žπ‘+𝑏2.

Here, we have a negative coefficient of the middle term, so the perfect trinomial will be in the form π‘Ž2βˆ’2π‘Žπ‘+𝑏2.

Therefore, we compare βˆ’60π‘₯2+25 and π‘Ž2βˆ’2π‘Žπ‘+𝑏2.

Equating the last terms, we have 𝑏2=25.

So, 𝑏=5.

Next, we can equate the middle terms, giving us βˆ’2π‘Žπ‘=βˆ’60π‘₯22π‘Žπ‘=60π‘₯2.

We can substitute in 𝑏=5 and simplify, giving us 2π‘Žπ‘=60π‘₯22π‘Ž(5)=60π‘₯210π‘Ž=60π‘₯2.

Dividing both sides by 10 gives us π‘Ž=6π‘₯2.

Therefore, π‘Ž2=36π‘₯4.

So, the completed expression to make a perfect square is 36π‘₯4βˆ’60π‘₯2+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)2βˆ’4.2Γ—10.1+(2.1)2.

Answer

We can use the properties of a perfect trinomial to solve this problem. Recall that π‘Ž2βˆ’2π‘Žπ‘+𝑏2=(π‘Žβˆ’π‘)2.

With π‘Ž=10.1 and 𝑏=2.1, we can write the given expression in the perfect trinomial form, π‘Ž2βˆ’2π‘Žπ‘+𝑏2, giving us (10.1)2βˆ’4.2Γ—10.1+(2.1)2=(10.1)2βˆ’2Γ—(10.1)(2.1)+(2.1)2.

Putting the right-hand side into the form (π‘Žβˆ’π‘)2 gives us (10.1)2βˆ’4.2Γ—10.1+(2.1)2=(10.1βˆ’2.1)2.

Simplifying and evaluating the right-hand side, we have (10.1)2βˆ’4.2Γ—10.1+(2.1)2=82=64.

Therefore, (10.1)2βˆ’4.2Γ—10.1+(2.1)2=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 81𝑐4+54𝑐2𝑑+9𝑑2. What is the length of its side?

Answer

We can use the properties of a perfect trinomial to solve this problem. Recall that π‘Ž2+2π‘Žπ‘+𝑏2=(π‘Ž+𝑏)2.

By setting π‘Ž=9𝑐2 and 𝑏=3𝑑, we can write the given expression in the perfect trinomial form π‘Ž2+2π‘Žπ‘+𝑏2, giving us 81𝑐4+54𝑐2𝑑+9𝑑2=ο€Ή9𝑐22+2Γ—ο€Ή9𝑐2(3𝑑)+(3𝑑)2.

Putting the right-hand side of the equation into the form (π‘Ž+𝑏)2 gives us 81𝑐4+54𝑐2𝑑+9𝑑2=ο€Ή9𝑐2+3𝑑2.

Therefore, since the area of a square is given by its side length squared, we can say that the length of the side is ||9𝑐2+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 π‘Ž2Β±2π‘Žπ‘+𝑏2 and can be factored into either (π‘Ž+𝑏)2 or (π‘Žβˆ’π‘)2, depending on the sign of the middle term, 2π‘Žπ‘, in the trinomial

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