# 7.4 Factor special products  (Page 2/7)

 Page 2 / 7

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

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

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

Factor: $64{y}^{2}-80y+25$ .

${\left(8y-5\right)}^{2}$

Factor: $16{z}^{2}-72z+81$ .

${\left(4z-9\right)}^{2}$

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

Factor: $36{x}^{2}+84xy+49{y}^{2}$ .

## Solution

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

Factor: $49{x}^{2}+84xy+36{y}^{2}$ .

${\left(7x+6y\right)}^{2}$

Factor: $64{m}^{2}+112mn+49{n}^{2}$ .

${\left(8m+7n\right)}^{2}$

Factor: $9{x}^{2}+50x+25$ .

## Solution

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

Factor: $16{r}^{2}+30rs+9{s}^{2}$ .

$\left(8r+3s\right)\left(2r+3s\right)$

Factor: $9{u}^{2}+87u+100$ .

$\left(3u+4\right)\left(3u+25\right)$

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.

Factor: $36{x}^{2}y-48xy+16y$ .

## Solution

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

Factor: $8{x}^{2}y-24xy+18y$ .

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

Factor: $27{p}^{2}q+90pq+75q$ .

$3q{\left(3p+5\right)}^{2}$

## Factor differences of squares

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

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

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.

## Product of conjugates pattern

If a and b are real numbers

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

The product is called a difference of squares.

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.

## Difference of squares pattern

If a and b are real numbers,

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.

## How to factor differences of squares

Factor: ${x}^{2}-4$ .

## Solution

Factor: ${h}^{2}-81$ .

$\left(h-9\right)\left(h+9\right)$

Factor: ${k}^{2}-121$ .

$\left(k-11\right)\left(k+11\right)$

## Factor differences of squares.

$\begin{array}{cccc}\mathbf{\text{Step 1.}}\phantom{\rule{0.2em}{0ex}}\text{Does the binomial fit the pattern?}\hfill & & & \hfill {a}^{2}-{b}^{2}\hfill \\ \phantom{\rule{2.5em}{0ex}}•\phantom{\rule{0.5em}{0ex}}\text{Is this a difference?}\hfill & & & \hfill ____-____\hfill \\ \phantom{\rule{2.5em}{0ex}}•\phantom{\rule{0.5em}{0ex}}\text{Are the first and last terms perfect squares?}\hfill & & & \\ \mathbf{\text{Step 2.}}\phantom{\rule{0.2em}{0ex}}\text{Write them as squares.}\hfill & & & \hfill {\left(a\right)}^{2}-{\left(b\right)}^{2}\hfill \\ \mathbf{\text{Step 3.}}\phantom{\rule{0.2em}{0ex}}\text{Write the product of conjugates.}\hfill & & & \hfill \left(a-b\right)\left(a+b\right)\hfill \\ \mathbf{\text{Step 4.}}\phantom{\rule{0.2em}{0ex}}\text{Check by multiplying.}\hfill & & & \end{array}$

will every polynomial have finite number of multiples?
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
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