Factor Special Products

Some trinomials are perfect squares. They result from multiplying a binomial times itself. We squared a binomial using the Binomial Squares pattern in a previous chapter.

The trinomial \(9{x}^{2}+24x+16\) is called a perfect square trinomial. It is the square of the binomial \(3x+4.\)

In this chapter, 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 \(a{x}^{2}+bx+c.\) 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.

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 the product, \(2ab?\) If everything checks, you can easily write the factors.

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 \({(a-b)}^{2}.\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here’s an example:

A difference of squares factors to a product of conjugates.

Remember, “difference” refers to subtraction. So, to use this pattern you must make sure you have a binomial in which two squares are being subtracted.

It is important to remember that sums of squares do not factor into a product of binomials. There are no binomial factors that multiply together to get a sum of squares. After removing any GCF, the expression \({a}^{2}+{b}^{2}\) is prime!

The next example shows variables in both terms.

As always, you should look for a common factor first whenever you have an expression to factor. Sometimes a common factor may “disguise” the difference of squares and you won’t recognize the perfect squares until you factor the GCF.

Also, to completely factor the binomial in the next example, we’ll factor a difference of squares twice!

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

There is another special pattern for factoring, one that we did not use when we multiplied polynomials. This is the pattern for the sum and difference of cubes. We will write these formulas first and then check them by multiplication.

We’ll check the first pattern and leave the second to you.

The two patterns look very similar, don’t they? But notice the signs in the factors. The sign of the binomial factor matches the sign in the original binomial. And the sign of the middle term of the trinomial factor is the opposite of the sign in the original binomial. If you recognize the pattern of the signs, it may help you memorize the patterns.

The trinomial factor in the sum and difference of cubes pattern cannot be factored.

It will be very helpful if you learn to recognize the cubes of the integers from 1 to 10, just like you have learned to recognize squares. We have listed the cubes of the integers from 1 to 10 in .

In the next example, we first factor out the GCF. Then we can recognize the sum of cubes.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

• Perfect Square Trinomials Pattern: If a and b are real numbers,
  \[\begin{array}{l}{a}^{2}+2ab+{b}^{2}={(a+b)}^{2} \\ {a}^{2}-2ab+{b}^{2}={(a-b)}^{2}\end{array}\]
• How to factor perfect square trinomials.
  \(\begin{array}{llllllll}\text{Step 1.} & \text{Does the trinomial fit the pattern?} & & & \ {a}^{2}+2ab+{b}^{2} & & & {a}^{2}-2ab+{b}^{2} \\ & \text{Is the first term a perfect square?} & & & \ {(a)}^{2} & & & {(a)}^{2} \\ & \text{Write it as a square.} & & & & & & \\ & \text{Is the last term a perfect square?} & & & \ {(a)}^{2}\ {(b)}^{2} & & & {(a)}^{2}\ {(b)}^{2} \\ & \text{Write it as a square.} & & & & & & \\ & \text{Check the middle term. Is it}\ 2ab? & & & \ {(a)}^{2}{}_{\text{↘}}\underset{2\cdot a\cdot b}{}{}_{\text{↙}}{(b)}^{2} & & & {(a)}^{2}{}_{\text{↘}}\underset{2\cdot a\cdot b}{}{}_{\text{↙}}{(b)}^{2} \\ \text{Step 2.} & \text{Write the square of the binomial.} & & & \ {(a+b)}^{2} & & & {(a-b)}^{2} \\ \text{Step 3.} & \text{Check by multiplying.} & & & & & & \end{array}\)
• Difference of Squares Pattern: If \(a,b\) are real numbers,
  
• How to factor differences of squares.
  \(\begin{array}{lllll}\text{Step 1.} & \text{Does the binomial fit the pattern?} & & & \ {a}^{2}-{b}^{2} \\ & \text{Is this a difference?} & & & \ \text{____}-\text{____} \\ & \text{Are the first and last terms perfect squares?} & & & \\ \text{Step 2.} & \text{Write them as squares.} & & & \ {(a)}^{2}-{(b)}^{2} \\ \text{Step 3.} & \text{Write the product of conjugates.} & & & \ (a-b)(a+b) \\ \text{Step 4.} & \text{Check by multiplying.} & & & \end{array}\)
• Sum and Difference of Cubes Pattern
  \(\begin{array}{l}{a}^{3}+{b}^{3}=(a+b)({a}^{2}-ab+{b}^{2}) \\ {a}^{3}-{b}^{3}=(a-b)({a}^{2}+ab+{b}^{2})\end{array}\)
• How to factor the sum or difference of cubes.
  1. Does the binomial fit the sum or difference of cubes pattern?
     Is it a sum or difference?
     Are the first and last terms perfect cubes?
  2. Write them as cubes.
  3. Use either the sum or difference of cubes pattern.
  4. Simplify inside the parentheses
  5. Check by multiplying the factors.

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

Mixed Practice

In the following exercises, factor completely.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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 chapter saves you a step. Let’s review the Binomial Squares pattern by squaring a binomial using FOIL.

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.

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.

When you square a binomial, the product is a perfect square trinomial. In this chapter, 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.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here’s an example:

Remember, when you multiply conjugate binomials, the middle terms of the product add to 0. All you have left is a binomial, the difference of squares.

Multiplying conjugates is the only way to get a binomial from the product of two binomials.

To factor, we will use the product pattern “in reverse” to factor the difference of squares. A difference of squares factors to a product of conjugates.

Remember, “difference” refers to subtraction. So, to use this pattern you must make sure you have a binomial in which two squares are being subtracted.

It is important to remember that sums of squares do not factor into a product of binomials. There are no binomial factors that multiply together to get a sum of squares. After removing any GCF, the expression \({a}^{2}+{b}^{2}\) is prime!

Condensed — the full section is in OpenStax Elementary Algebra 2e.

There is another special pattern for factoring, one that we did not use when we multiplied polynomials. This is the pattern for the sum and difference of cubes. We will write these formulas first and then check them by multiplication.

We’ll check the first pattern and leave the second to you.

The two patterns look very similar, don’t they? But notice the signs in the factors. The sign of the binomial factor matches the sign in the original binomial. And the sign of the middle term of the trinomial factor is the opposite of the sign in the original binomial. If you recognize the pattern of the signs, it may help you memorize the patterns.

The trinomial factor in the sum and difference of cubes pattern cannot be factored.

It can be very helpful if you learn to recognize the cubes of the integers from 1 to 10, just like you have learned to recognize squares. We have listed the cubes of the integers from 1 to 10 in .

Be careful to use the correct signs in the factors of the sum and difference of cubes.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

• Factor perfect square trinomials See .
  \(\begin{array}{lllllll}\text{Step 1.}\ \text{Does the trinomial fit the pattern?} & & & {a}^{2}+2ab+{b}^{2} & & & {a}^{2}-2ab+{b}^{2} \\ \text{Is the first term a perfect square?} & & & {(a)}^{2} & & & {(a)}^{2} \\ \text{Write it as a square.} & & & & & & \\ \text{Is the last term a perfect square?} & & & {(a)}^{2}\ {(b)}^{2} & & & {(a)}^{2}\ {(b)}^{2} \\ \text{Write it as a square.} & & & & & & \\ \text{Check the middle term. Is it}\ 2ab? & & & {(a)}^{2}{}_{\text{↘}}\underset{2\cdot a\cdot b}{}{}_{\text{↙}}{(b)}^{2} & & & {(a)}^{2}{}_{\text{↘}}\underset{2\cdot a\cdot b}{}{}_{\text{↙}}{(b)}^{2} \\ \text{Step 2.}\ \text{Write the square of the binomial.} & & & {(a+b)}^{2} & & & {(a-b)}^{2} \\ \text{Step 3.}\ \text{Check by multiplying.} & & & & & & \end{array}\)
• Factor differences of squares See .
  \(\begin{array}{llll}\text{Step 1.}\ \text{Does the binomial fit the pattern?} & & & {a}^{2}-{b}^{2} \\ \text{Is this a difference?} & & & ____-____ \\ \text{Are the first and last terms perfect squares?} & & & \\ \text{Step 2.}\ \text{Write them as squares.} & & & {(a)}^{2}-{(b)}^{2} \\ \text{Step 3.}\ \text{Write the product of conjugates.} & & & (a-b)(a+b) \\ \text{Step 4.}\ \text{Check by multiplying.} & & & \end{array}\)
• Factor sum and difference of cubes To factor the sum or difference of cubes: See .
  1. Does the binomial fit the sum or difference of cubes pattern? Is it a sum or difference? Are the first and last terms perfect cubes?
  2. Write them as cubes.
  3. Use either the sum or difference of cubes pattern.
  4. Simplify inside the parentheses
  5. Check by multiplying the factors.

Factor Perfect Square Trinomials

In the following exercises, factor.

Factor Differences of Squares

In the following exercises, factor.

Factor Sums and Differences of Cubes

In the following exercises, factor.

Mixed Practice

In the following exercises, factor.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Condensed — the full section is in OpenStax Elementary Algebra 2e.