# 7.4 Factor special products

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By the end of this section, you will be able to:
• Factor perfect square trinomials
• Factor differences of squares
• Factor sums and differences of cubes
• Choose method to factor a polynomial completely

Before you get started, take this readiness quiz.

1. Simplify: ${\left(12x\right)}^{2}.$
If you missed this problem, review [link] .
2. Multiply: ${\left(m+4\right)}^{2}.$
If you missed this problem, review [link] .
3. Multiply: ${\left(p-9\right)}^{2}.$
If you missed this problem, review [link] .
4. Multiply: $\left(k+3\right)\left(k-3\right).$
If you missed this problem, review [link] .

The strategy for factoring we developed in the last section 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.

## Factor 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 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.

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

The trinomial 9 x 2 + 24 +16 is called a perfect square trinomial. It is the square of the binomial 3 x +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,

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

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 ax 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.

## Perfect square trinomials pattern

If a and b are real numbers,

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

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, 2 ab ? If everything checks, you can easily write the factors.

## How to factor perfect square trinomials

Factor: $9{x}^{2}+12x+4$ .

## Solution

Factor: $4{x}^{2}+12x+9$ .

${\left(2x+3\right)}^{2}$

Factor: $9{y}^{2}+24y+16$ .

${\left(3y+4\right)}^{2}$

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 ${\left(a-b\right)}^{2}$ .

The steps are summarized here.

## Factor perfect square trinomials.

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

#### Questions & Answers

a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost$3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use? Jake Reply enrique borrowed$23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total$23,500
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost$20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be \$10 per square foot?
The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water.
Bridget
help me understand graphs
what kind of graphs?
bruce
function f(x) to find each value
Marlene
I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much.
Marlene
Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6
Melissa
if they say find f(5) then the answer would be f(5)=12
Melissa
I need you to help me Melissa. Wish I can show you my homework
Marlene
How is f(1) =0 I am really confused
Marlene
what's the formula given? f(x)=?
Melissa
It shows a graph that I wish I could send photo of to you on here
Marlene
Which problem specifically?
Melissa
which problem?
Melissa
I don't know any to be honest. But whatever you can help me with for I can practice will help
Marlene
I got it. sorry, was out and about. I'll look at it now.
Melissa
Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things.
Marlene
I just responded.
Melissa
Thank you
Marlene
-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
State the question clearly please
Rich
write in this form a/b answer should be in the simplest form 5%
convert to decimal 9/11
August
0.81818
Rich
5/100 = .05 but Rich is right that 9/11 = .81818
Melissa
Equation in the form of a pending point y+2=1/6(×-4)
write in simplest form 3 4/2
August