# 7.5 General strategy for factoring polynomials  (Page 2/2)

 Page 2 / 2

Factor completely: $16{x}^{3}-36x$ .

$4x\left(2x-3\right)\left(2x+3\right)$

Factor completely: $27{y}^{2}-48$ .

$3\left(3y-4\right)\left(3y+4\right)$

Factor completely: $4{a}^{2}-12ab+9{b}^{2}$ .

## Solution

 Is there a GCF? No. Is it a binomial, trinomial, or are there more terms? Trinomial with $a\ne 1$ . But the first term is a   perfect square. Is the last term a perfect square? Yes. Does it fit the pattern, ${a}^{2}-2ab+{b}^{2}?$ Yes. Write it as a square. Check your answer. Is the expression factored completely? Yes. The binomial is not a difference of squares. Multiply. ${\left(2a-3b\right)}^{2}$ ${\left(2a\right)}^{2}-2\cdot 2a\cdot 3b+{\left(3b\right)}^{2}$ $4{a}^{2}-12ab+9{b}^{2}✓$

Factor completely: $4{x}^{2}+20xy+25{y}^{2}$ .

${\left(2x+5y\right)}^{2}$

Factor completely: $9{m}^{2}+42mn+49{n}^{2}$ .

${\left(3m+7n\right)}^{2}$

Factor completely: $6{y}^{2}-18y-60$ .

## Solution

$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 6.}\hfill & & & \hfill 6{y}^{2}-18y-60\hfill \\ \text{Factor out the GCF.}\hfill & & & \text{Trinomial with leading coefficient 1.}\hfill & & & \hfill 6\left({y}^{2}-3y-10\right)\hfill \\ \begin{array}{c}\text{In the parentheses, is it a binomial, trinomial,}\hfill \\ \text{or are there more terms?}\hfill \end{array}\hfill & & & & & & \\ \text{“Undo” FOIL.}\hfill & & & \hfill 6\left(y\phantom{\rule{1em}{0ex}}\right)\left(y\phantom{\rule{1em}{0ex}}\right)\hfill & & & \hfill 6\left(y+2\right)\left(y-5\right)\hfill \\ \\ \\ \text{Check your answer.}\hfill & & & & & & \\ \text{Is the expression factored completely?}\hfill & & & & & & \hfill \text{Yes.}\hfill \\ \text{Neither binomial is a difference of squares.}\hfill & & & & & & \\ \text{Multiply.}\hfill & & & & & & \\ \\ \\ \phantom{\rule{2.5em}{0ex}}6\left(y+2\right)\left(y-5\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}6\left({y}^{2}-5y+2y-10\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}6\left({y}^{2}-3y-10\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}6{y}^{2}-18y-60\phantom{\rule{0.2em}{0ex}}✓\hfill & & & \end{array}$

Factor completely: $8{y}^{2}+16y-24$ .

$8\left(y-1\right)\left(y+3\right)$

Factor completely: $5{u}^{2}-15u-270$ .

$5\left(u-9\right)\left(u+6\right)$

Factor completely: $24{x}^{3}+81$ .

## Solution

 Is there a GCF? Yes, 3. $24{x}^{3}+81$ Factor it out. $3\left(8{x}^{3}+27\right)$ In the parentheses, is it a binomial, trinomial, or are there more than three terms? Binomial. Is it a sum or difference? Sum. Of squares or cubes? Sum of cubes. Write it using the sum of cubes pattern. Is the expression factored completely? Yes. $3\left(2x+3\right)\left(4{x}^{2}-6x+9\right)$ Check by multiplying. We leave the check to you.

Factor completely: $250{m}^{3}+432$ .

$2\left(5m+6\right)\left(25{m}^{2}-30m+36\right)$

Factor completely: $81{q}^{3}+192$ .

$81\left(q+2\right)\left({q}^{2}-2q+4\right)$

Factor completely: $2{x}^{4}-32$ .

## Solution

$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 2.}\hfill & & & \hfill 2{x}^{4}-32\hfill \\ \text{Factor it out.}\hfill & & & & & & \hfill 2\left({x}^{4}-16\right)\hfill \\ \text{In the parentheses, is it a binomial, trinomial,}\hfill & & & & & & \\ \text{or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Is it a sum or difference?}\hfill & & & \text{Yes.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Of squares or cubes?}\hfill & & & \text{Difference of squares.}\hfill & & & \hfill 2\left({\left({x}^{2}\right)}^{2}-{\left(4\right)}^{2}\right)\hfill \\ \text{Write it as a product of conjugates.}\hfill & & & & & & \hfill 2\left({x}^{2}-4\right)\left({x}^{2}+4\right)\hfill \\ \text{The first binomial is again a difference of squares.}\hfill & & & & & & \hfill 2\left({\left(x\right)}^{2}-{\left(2\right)}^{2}\right)\left({x}^{2}+4\right)\hfill \\ \text{Write it as a product of conjugates.}\hfill & & & & & & \hfill 2\left(x-2\right)\left(x+2\right)\left({x}^{2}+4\right)\hfill \\ \text{Is the expression factored completely?}\hfill & & & \text{Yes.}\hfill & & & \\ \\ \\ \phantom{\rule{1em}{0ex}}\text{None of these binomials is a difference of squares.}\hfill & & & & & & \\ \text{Check your answer.}\hfill & & & & & & \\ \\ \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \\ \\ \\ \\ \\ \phantom{\rule{3em}{0ex}}\begin{array}{c}2\left(x-2\right)\left(x+2\right)\left({x}^{2}+4\right)\hfill \\ 2\left({x}^{2}-4\right)\left({x}^{2}+4\right)\hfill \\ 2\left({x}^{4}-16\right)\hfill \\ 2{x}^{4}-32✓\hfill \end{array}\hfill & & & & & & \end{array}$

Factor completely: $4{a}^{4}-64$ .

$4\left({a}^{2}+4\right)\left(a-2\right)\left(a+2\right)$

Factor completely: $7{y}^{4}-7$ .

$7\left({y}^{2}+1\right)\left(y-1\right)\left(y+1\right)$

Factor completely: $3{x}^{2}+6bx-3ax-6ab$ .

## Solution

$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 3.}\hfill & & & \hfill 3{x}^{2}+6bx-3ax-6ab\hfill \\ \\ \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill 3\left({x}^{2}+2bx-ax-2ab\right)\hfill \\ \\ \\ \text{In the parentheses, is it a binomial, trinomial,}\hfill & & & \text{More than 3}\hfill & & & \\ \text{or are there more terms?}\hfill & & & \text{terms.}\hfill & & & \\ \\ \\ \text{Use grouping.}\hfill & & & & & & \hfill 3\left[x\left(x+2b\right)-a\left(x+2b\right)\right]\hfill \\ & & & & & & \hfill 3\left(x+2b\right)\left(x-a\right)\hfill \\ \text{Check your answer.}\hfill & & & & & & \\ \\ \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely? Yes.}\hfill & & & & & & \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}3\left(x+2b\right)\left(x-a\right)\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}3\left({x}^{2}-ax+2bx-2ab\right)\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}3{x}^{2}-3ax+6bx-6ab\phantom{\rule{0.2em}{0ex}}✓\hfill & & & & & & \end{array}$

Factor completely: $6{x}^{2}-12xc+6bx-12bc$ .

$6\left(x+b\right)\left(x-2c\right)$

Factor completely: $16{x}^{2}+24xy-4x-6y$ .

$2\left(4x-1\right)\left(x+3y\right)$

Factor completely: $10{x}^{2}-34x-24$ .

## Solution

$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 2.}\hfill & & & \hfill 10{x}^{2}-34x-24\hfill \\ \\ \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill 2\left(5{x}^{2}-17x-12\right)\hfill \\ \\ \\ \text{In the parentheses, is it a binomial, trinomial,}\hfill & & & \hfill \text{Trinomial with}\hfill & & & \\ \text{or are there more than three terms?}\hfill & & & a\ne 1.\hfill & & & \\ \\ \\ \text{Use trial and error or the “ac” method.}\hfill & & & & & & \hfill 2\underset{}{\left(5{x}^{2}}-17x\underset{}{-12\right)}\hfill \\ & & & & & & \hfill 2\left(5x+3\right)\left(x-4\right)\hfill \\ \\ \\ \begin{array}{c}\text{Check your answer. Is the expression factored}\hfill \\ \text{completely? Yes.}\hfill \end{array}\hfill & & & & & & \\ \\ \\ \phantom{\rule{2em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}2\left(5x+3\right)\left(x-4\right)\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}2\left(5{x}^{2}-20x+3x-12\right)\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}2\left(5{x}^{2}-17x-12\right)\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}10{x}^{2}-34x-24\phantom{\rule{0.2em}{0ex}}✓\hfill & & & & & & \end{array}$

Factor completely: $4{p}^{2}-16p+12$ .

$4\left(p-1\right)\left(p-3\right)$

Factor completely: $6{q}^{2}-9q-6$ .

$3\left(q-2\right)\left(2q+1\right)$

## Key concepts

• General Strategy for Factoring Polynomials See [link] .
• How to Factor Polynomials
1. Is there a greatest common factor? Factor it out.
2. Is the polynomial a binomial, trinomial, or are there more than three terms?
• If it is a binomial:
Is it a sum?
• Of squares? Sums of squares do not factor.
• Of cubes? Use the sum of cubes pattern.
Is it a difference?
• Of squares? Factor as the product of conjugates.
• Of cubes? Use the difference of cubes pattern.
• If it is a trinomial:
Is it of the form ${x}^{2}+bx+c$ ? Undo FOIL.
Is it of the form $a{x}^{2}+bx+c$ ?
• If ‘a’ and ‘c’ are squares, check if it fits the trinomial square pattern.
• Use the trial and error or ‘ac’ method.
• If it has more than three terms:
Use the grouping method.
3. Check. Is it factored completely? Do the factors multiply back to the original polynomial?

## Practice makes perfect

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$10{x}^{4}+35{x}^{3}$

$5{x}^{3}\left(2x+7\right)$

$18{p}^{6}+24{p}^{3}$

${y}^{2}+10y-39$

$\left(y-3\right)\left(y+13\right)$

${b}^{2}-17b+60$

$2{n}^{2}+13n-7$

$\left(2n-1\right)\left(n+7\right)$

$8{x}^{2}-9x-3$

${a}^{5}+9{a}^{3}$

${a}^{3}\left({a}^{2}+9\right)$

$75{m}^{3}+12m$

$121{r}^{2}-{s}^{2}$

$\left(11r-s\right)\left(11r+s\right)$

$49{b}^{2}-36{a}^{2}$

$8{m}^{2}-32$

$8\left(m-2\right)\left(m+2\right)$

$36{q}^{2}-100$

$25{w}^{2}-60w+36$

${\left(5w-6\right)}^{2}$

$49{b}^{2}-112b+64$

${m}^{2}+14mn+49{n}^{2}$

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

$64{x}^{2}+16xy+{y}^{2}$

$7{b}^{2}+7b-42$

$7\left(b+3\right)\left(b-2\right)$

$3{n}^{2}+30n+72$

$3{x}^{3}-81$

$3\left(x-3\right)\left({x}^{2}+3x+9\right)$

$5{t}^{3}-40$

${k}^{4}-16$

$\left(k-2\right)\left(k+2\right)\left({k}^{2}+4\right)$

${m}^{4}-81$

$15pq-15p+12q-12$

$3\left(5p+4\right)\left(q-1\right)$

$12ab-6a+10b-5$

$4{x}^{2}+40x+84$

$4\left(x+3\right)\left(x+7\right)$

$5{q}^{2}-15q-90$

${u}^{5}+{u}^{2}$

${u}^{2}\left(u+1\right)\left({u}^{2}-u+1\right)$

$5{n}^{3}+320$

$4{c}^{2}+20cd+81{d}^{2}$

prime

$25{x}^{2}+35xy+49{y}^{2}$

$10{m}^{4}-6250$

$10\left(m-5\right)\left(m+5\right)\left({m}^{2}+25\right)$

$3{v}^{4}-768$

## Everyday math

Watermelon drop A springtime tradition at the University of California San Diego is the Watermelon Drop, where a watermelon is dropped from the seventh story of Urey Hall.

1. The binomial $-16{t}^{2}+80$ gives the height of the watermelon $t$ seconds after it is dropped. Factor the greatest common factor from this binomial.
2. If the watermelon is thrown down with initial velocity 8 feet per second, its height after $t$ seconds is given by the trinomial $-16{t}^{2}-8t+80$ . Completely factor this trinomial.

$-16\left({t}^{2}-5\right)$ $-8\left(2t+5\right)\left(t-2\right)$

Pumpkin drop A fall tradition at the University of California San Diego is the Pumpkin Drop, where a pumpkin is dropped from the eleventh story of Tioga Hall.

1. The binomial $-16{t}^{2}+128$ gives the height of the pumpkin t seconds after it is dropped. Factor the greatest common factor from this binomial.
2. If the pumpkin is thrown down with initial velocity 32 feet per second, its height after $t$ seconds is given by the trinomial $-16{t}^{2}-32t+128$ . Completely factor this trinomial.

## Writing exercises

The difference of squares ${y}^{4}-625$ can be factored as $\left({y}^{2}-25\right)\left({y}^{2}+25\right)$ . But it is not completely factored. What more must be done to completely factor it?

Of all the factoring methods covered in this chapter (GCF, grouping, undo FOIL, ‘ac’ method, special products) which is the easiest for you? Which is the hardest? Explain your answers.

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
What does 3 times your weight right now
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
(×+1)(×-4) = x^2-3×-4
Gee
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs$4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon? Mohamed Reply (a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r) muhammad Reply 4x-7y=8 2x-7y=1 what is the answer? Ramil Reply x=7/2 & y=6/7 Pbp x=7/2 & y=6/7 use Elimination Debra true bismark factoriz e usman 4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2 Peggie Ok cool answer peggie Frank thanks Ramil copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x Sandra Reply What is c+4=8 Penny Reply 2 Letha 4 Lolita 4 Rich 4 thinking C+4=8 -4 -4 C =4 thinking I need to study Letha 4+4=8 William During two years in college, a student earned$9,500. The second year, she earned \$500 more than twice the amount she earned the first year.
9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
Pbp  By By Eric Crawford       By