Factorising Perfect Square Trinomials

The strategy for factoring or factorising or factorizing we developed in the last topic will guide you as you factor most binomials, trinomials, and polynomials with more than three terms. We have seen that some binomials and trinomials result from special products—squaring binomials and multiplying conjugates. If you learn to recognize these kinds of polynomials, you can use the special products patterns to factor them much more quickly.

Factorising Perfect Square Trinomials

Some trinomials are perfect squares. They result from multiplying a binomial times itself. You can square a binomial by using FOIL, but using the Binomial Squares pattern you saw in a previous tutorial saves you a step. Let’s review the Binomial Squares pattern by squaring a binomial using FOIL.

This image shows the FOIL procedure for multiplying (3x + 4) squared. The polynomial is written with two factors (3x + 4)(3x + 4). Then, the terms are 9 x squared + 12 x + 12 x + 16, demonstrating first, outer, inner, last. Finally, the product is written, 9 x squared + 24 x + 16.

The first term is the square of the first term of the binomial and the last term is the square of the last. The middle term is twice the product of the two terms of the binomial.

\(\begin{array}{c}\hfill {(3x)}^{2}+2(3x·4)+{4}^{2}\hfill \\ \hfill 9{x}^{2}+24x+16\hfill \end{array}\)

The trinomial 9x² + 24 +16 is called a perfect square trinomial. It is the square of the binomial 3x+4.

We’ll repeat the Binomial Squares Pattern here to use as a reference in factoring.

Binomial Squares Pattern

If a and b are real numbers,

\(\begin{array}{cccc}\hfill {(a+b)}^{2}={a}^{2}+2ab+{b}^{2}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}{(a-b)}^{2}={a}^{2}-2ab+{b}^{2}\hfill \end{array}\)

When you square a binomial, the product is a perfect square trinomial. In this tutorial, you are learning to factor—now, you will start with a perfect square trinomial and factor it into its prime factors.

You could factor this trinomial using the methods described in the last section, since it is of the form ax² + bx + c. But if you recognize that the first and last terms are squares and the trinomial fits the perfect square trinomials pattern, you will save yourself a lot of work.

Here is the pattern—the reverse of the binomial squares pattern.

Perfect Square Trinomials Pattern

If a and b are real numbers,

\(\begin{array}{cccc}\hfill {a}^{2}+2ab+{b}^{2}={(a+b)}^{2}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}{a}^{2}-2ab+{b}^{2}={(a-b)}^{2}\hfill \end{array}\)

To make use of this pattern, you have to recognize that a given trinomial fits it. Check first to see if the leading coefficient is a perfect square, \({a}^{2}\). Next check that the last term is a perfect square, \({b}^{2}\). Then check the middle term—is it twice the product, 2ab? If everything checks, you can easily write the factors.

Example: How to Factor Perfect Square Trinomials

Factor: \(9{x}^{2}+12x+4\).

Solution

This table gives the steps for factoring 9 x squared +12 x +4. The first step is recognizing the perfect square pattern “a” squared + 2 a b + b squared. This includes, is the first term a perfect square and is the last term a perfect square. The first term can be written as (3 x) squared and the last term can be written as 2 squared. Also, in the first step, the middle term has to be twice “a” times b. This is verified by 2 times 3 x times 2 being 12 x.

The second step is writing the square of the binomial. The polynomial is written as (3 x) squared + 2 times 3 x times 2 + 2 squared. This is factored as (3 x + 2) squared.

The last step is to check with multiplication.

The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern \({a}^{2}-2ab+{b}^{2}\), which factors to \({(a-b)}^{2}\).

The steps are summarized here.

Factor perfect square trinomials.

\(\begin{array}{ccccccc}\mathbf{\text{Step 1.}}\phantom{\rule{0.2em}{0ex}}\text{Does the trinomial fit the pattern?}\hfill & & & \hfill {a}^{2}+2ab+{b}^{2}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}{a}^{2}-2ab+{b}^{2}\hfill \\ \phantom{\rule{2.5em}{0ex}}•\phantom{\rule{0.5em}{0ex}}\text{Is the first term a perfect square?}\hfill & & & \hfill {(a)}^{2}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}{(a)}^{2}\hfill \\ \phantom{\rule{4em}{0ex}}\text{Write it as a square.}\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}•\phantom{\rule{0.5em}{0ex}}\text{Is the last term a perfect square?}\hfill & & & {(a)}^{2}\phantom{\rule{4.5em}{0ex}}{(b)}^{2}\hfill & & & \phantom{\rule{2em}{0ex}}{(a)}^{2}\phantom{\rule{4.5em}{0ex}}{(b)}^{2}\hfill \\ \phantom{\rule{4em}{0ex}}\text{Write it as a square.}\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}•\phantom{\rule{0.5em}{0ex}}\text{Check the middle term. Is it}\phantom{\rule{0.2em}{0ex}}2ab?\hfill & & & {(a)}^{2}{}_{\text{↘}}\underset{2·a·b}{}{}_{\text{↙}}{(b)}^{2}\hfill & & & \phantom{\rule{2em}{0ex}}{(a)}^{2}{}_{\text{↘}}\underset{2·a·b}{}{}_{\text{↙}}{(b)}^{2}\hfill \\ \mathbf{\text{Step 2.}}\phantom{\rule{0.2em}{0ex}}\text{Write the square of the binomial.}\hfill & & & \hfill {(a+b)}^{2}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}{(a-b)}^{2}\hfill \\ \mathbf{\text{Step 3.}}\phantom{\rule{0.2em}{0ex}}\text{Check by multiplying.}\hfill & & & & & & \end{array}\)

We’ll work one now where the middle term is negative.

Example

Factor: \(81{y}^{2}-72y+16\).

Solution

The first and last terms are squares. See if the middle term fits the pattern of a perfect square trinomial. The middle term is negative, so the binomial square would be \({(a-b)}^{2}\).


Are the first and last terms perfect squares?
Check the middle term.
Does is match \({(a-b)}^{2}\)? Yes.
Write the square of a binomial.
Check by mulitplying.
\({(9y-4)}^{2}\)
\({(9y)}^{2}-2\cdot 9y\cdot 4+{4}^{2}\)
\(81{y}^{2}-72y+16✓\)

The next example will be a perfect square trinomial with two variables.

Example

Factor: \(36{x}^{2}+84xy+49{y}^{2}\).

Solution


Test each term to verify the pattern.
Factor.
Check by mulitplying.
\({(6x+7y)}^{2}\)
\({(6x)}^{2}+2\cdot 6x\cdot 7y+{(7y)}^{2}\)
\(36{x}^{2}+84xy+49{y}^{2}✓\)

Example

Factor: \(9{x}^{2}+50x+25\).

Solution

\(\begin{array}{cccc}& & & \hfill 9{x}^{2}+50x+25\hfill \\ \text{Are the first and last terms perfect squares?}\hfill & & & \hfill {(3x)}^{2}\phantom{\rule{3em}{0ex}}{(5)}^{2}\hfill \\ \text{Check the middle term—is it}\phantom{\rule{0.2em}{0ex}}2ab?\hfill & & & \hfill {(3x)}^{2}{}_{\text{↘}}\underset{\underset{30x}{2(3x)(5)}}{\text{}}{}_{\text{↙}}{(5)}^{2}\hfill \\ \text{No!}\phantom{\rule{0.2em}{0ex}}30x\ne 50x\hfill & & & \text{This does not fit the pattern!}\hfill \\ \text{Factor using the “ac” method.}\hfill & & & \hfill 9{x}^{2}+50x+25\hfill \\ \\ \text{Notice:}\phantom{\rule{0.2em}{0ex}}\begin{array}{c}\hfill ac\hfill \\ \hfill 9·25\hfill \\ \hfill 225\hfill \end{array}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\begin{array}{c}\hfill 5·45=225\hfill \\ \hfill 5+45=50\hfill \end{array}\hfill \\ \text{Split the middle term.}\hfill & & & \hfill 9{x}^{2}+5x+45x+25\hfill \\ \text{Factor by grouping.}\hfill & & & \hfill x(9x+5)+5(9x+5)\hfill \\ & & & \hfill (9x+5)(x+5)\hfill \\ \text{Check.}\hfill & & & \\ \\ \phantom{\rule{2.5em}{0ex}}(9x+5)(x+5)\hfill & & & \\ \phantom{\rule{2.5em}{0ex}}9{x}^{2}+45x+5x+25\hfill & & & \\ \phantom{\rule{2.5em}{0ex}}9{x}^{2}+50x+25\phantom{\rule{0.2em}{0ex}}✓\hfill & & & \end{array}\)

Remember the very first step in our Strategy for Factoring Polynomials? It was to ask “is there a greatest common factor?” and, if there was, you factor the GCF before going any further. Perfect square trinomials may have a GCF in all three terms and it should be factored out first. And, sometimes, once the GCF has been factored, you will recognize a perfect square trinomial.

Example

Factor: \(36{x}^{2}y-48xy+16y\).

Solution

	\(36{x}^{2}y-48xy+16y\)
Is there a GCF? Yes, 4y, so factor it out.	\(4y(9{x}^{2}-12x+4)\)
Is this a perfect square trinomial?
Verify the pattern.
Factor.	\(4y{(3x-2)}^{2}\)
Remember: Keep the factor 4y in the final product.
Check.
\(4y{(3x-2)}^{2}\)
\(4y[{(3x)}^{2}-2·3x·2+{2}^{2}]\)
\(4y{(9x)}^{2}-12x+4\)
\(36{x}^{2}y-48xy+16y✓\)

Factorising Perfect Square Trinomials