# 7.1 Greatest common factor and factor by grouping  (Page 3/5)

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Factor: $-16z-64$ .

$-8\left(8z+8\right)$

Factor: $-9y-27$ .

$-9\left(y+3\right)$

Factor: $-6{a}^{2}+36a$ .

## Solution

The leading coefficient is negative, so the GCF will be negative.?

 Since the leading coefficient is negative, the GCF is negative, −6 a . Rewrite each term using the GCF. Factor the GCF. Check. $-6a\left(a-6\right)$ $-6a\cdot a+\left(-6a\right)\left(-6\right)$ $-6{a}^{2}+36a✓$

Factor: $-4{b}^{2}+16b$ .

$-4b\left(b-4\right)$

Factor: $-7{a}^{2}+21a$ .

$-7a\left(a-3\right)$

Factor: $5q\left(q+7\right)-6\left(q+7\right)$ .

## Solution

The GCF is the binomial $q+7$ .

 Factor the GCF, ( q + 7). Check on your own by multiplying.

Factor: $4m\left(m+3\right)-7\left(m+3\right)$ .

$\left(m+3\right)\left(4m-7\right)$

Factor: $8n\left(n-4\right)+5\left(n-4\right)$ .

$\left(n-4\right)\left(8n+5\right)$

## Factor by grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

(Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.)

## How to factor by grouping

Factor: $xy+3y+2x+6$ .

## Solution

Factor: $xy+8y+3x+24$ .

$\left(x+8\right)\left(y+3\right)$

Factor: $ab+7b+8a+56$ .

$\left(a+7\right)\left(b+8\right)$

## Factor by grouping.

1. Group terms with common factors.
2. Factor out the common factor in each group.
3. Factor the common factor from the expression.
4. Check by multiplying the factors.

Factor: ${x}^{2}+3x-2x-6$ .

## Solution

$\begin{array}{cccc}\text{There is no GCF in all four terms.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{x}^{2}+3x\phantom{\rule{0.5em}{0ex}}-2x-6\hfill \\ \text{Separate into two parts.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\underset{⎵}{{x}^{2}+3x}\phantom{\rule{0.5em}{0ex}}\underset{⎵}{-2x-6}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the GCF from both parts. Be careful}\hfill \\ \text{with the signs when factoring the GCF from}\hfill \\ \text{the last two terms.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\begin{array}{c}\hfill x\left(x+3\right)-2\left(x+3\right)\hfill \\ \hfill \left(x+3\right)\left(x-2\right)\hfill \end{array}\hfill \\ \\ \\ \text{Check on your own by multiplying.}\hfill & & & \end{array}$

Factor: ${x}^{2}+2x-5x-10$ .

$\left(x-5\right)\left(x+2\right)$

Factor: ${y}^{2}+4y-7y-28$ .

$\left(y+4\right)\left(y-7\right)$

Access these online resources for additional instruction and practice with greatest common factors (GFCs) and factoring by grouping.

## Key concepts

• Finding the Greatest Common Factor (GCF): To find the GCF of two expressions:
1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
2. List all factors—matching common factors in a column. In each column, circle the common factors.
3. Bring down the common factors that all expressions share.
4. Multiply the factors as in [link] .
• Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial:
1. Find the GCF of all the terms of the polynomial.
2. Rewrite each term as a product using the GCF.
3. Use the ‘reverse’ Distributive Property to factor the expression.
4. Check by multiplying the factors as in [link] .
• Factor by Grouping: To factor a polynomial with 4 four or more terms
1. Group terms with common factors.
2. Factor out the common factor in each group.
3. Factor the common factor from the expression.
4. Check by multiplying the factors as in [link] .

## Practice makes perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

8, 18

2

24, 40

72, 162

18

150, 275

10 a , 50

10

5 b , 30

$3x,10{x}^{2}$

$x$

$21{b}^{2},14b$

$8{w}^{2},24{w}^{3}$

$8{w}^{2}$

$30{x}^{2},18{x}^{3}$

$10{p}^{3}q,12p{q}^{2}$

$2pq$

$8{a}^{2}{b}^{3},10a{b}^{2}$

$12{m}^{2}{n}^{3},30{m}^{5}{n}^{3}$

$6{m}^{2}{n}^{3}$

$28{x}^{2}{y}^{4},42{x}^{4}{y}^{4}$

$10{a}^{3},12{a}^{2},14a$

$2a$

$20{y}^{3},28{y}^{2},40y$

$35{x}^{3},10{x}^{4},5{x}^{5}$

$5{x}^{3}$

$27{p}^{2},45{p}^{3},9{p}^{4}$

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$4x+20$

$4\left(x+5\right)$

$8y+16$

$6m+9$

$3\left(2m+3\right)$

$14p+35$

$9q+9$

$9\left(q+1\right)$

$7r+7$

$8m-8$

$8\left(m-1\right)$

$4n-4$

$9n-63$

$9\left(n-7\right)$

$45b-18$

$3{x}^{2}+6x-9$

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

$4{y}^{2}+8y-4$

$8{p}^{2}+4p+2$

$2\left(4{p}^{2}+2p+1\right)$

$10{q}^{2}+14q+20$

$8{y}^{3}+16{y}^{2}$

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

$12{x}^{3}-10x$

$5{x}^{3}-15{x}^{2}+20x$

$5x\left({x}^{2}-3x+4\right)$

$8{m}^{2}-40m+16$

$12x{y}^{2}+18{x}^{2}{y}^{2}-30{y}^{3}$

$6{y}^{2}\left(2x+3{x}^{2}-5y\right)$

$21p{q}^{2}+35{p}^{2}{q}^{2}-28{q}^{3}$

$-2x-4$

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

$-3b+12$

$5x\left(x+1\right)+3\left(x+1\right)$

$\left(x+1\right)\left(5x+3\right)$

$2x\left(x-1\right)+9\left(x-1\right)$

$3b\left(b-2\right)-13\left(b-2\right)$

$\left(b-2\right)\left(3b-13\right)$

$6m\left(m-5\right)-7\left(m-5\right)$

Factor by Grouping

In the following exercises, factor by grouping.

$xy+2y+3x+6$

$\left(y+3\right)\left(x+2\right)$

$mn+4n+6m+24$

$uv-9u+2v-18$

$\left(u+2\right)\left(v-9\right)$

$pq-10p+8q-80$

${b}^{2}+5b-4b-20$

$\left(b-4\right)\left(b+5\right)$

${m}^{2}+6m-12m-72$

${p}^{2}+4p-9p-36$

$\left(p-9\right)\left(p+4\right)$

${x}^{2}+5x-3x-15$

Mixed Practice

In the following exercises, factor.

$-20x-10$

$-10\left(2x+1\right)$

$5{x}^{3}-{x}^{2}+x$

$3{x}^{3}-7{x}^{2}+6x-14$

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

${x}^{3}+{x}^{2}-x-1$

${x}^{2}+xy+5x+5y$

$\left(x+y\right)\left(x+5\right)$

$5{x}^{3}-3{x}^{2}-5x-3$

## Everyday math

Area of a rectangle The area of a rectangle with length 6 less than the width is given by the expression ${w}^{2}-6w$ , where $w=$ width. Factor the greatest common factor from the polynomial.

$w\left(w-6\right)$

Height of a baseball The height of a baseball t seconds after it is hit is given by the expression $-16{t}^{2}+80t+4$ . Factor the greatest common factor from the polynomial.

## Writing exercises

The greatest common factor of 36 and 60 is 12. Explain what this means.

What is the GCF of ${y}^{4},{y}^{5},\text{and}\phantom{\rule{0.2em}{0ex}}{y}^{10}$ ? Write a general rule that tells you how to find the GCF of ${y}^{a},{y}^{b},\text{and}{y}^{c}$ .

## Self check

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

If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.
how do i set this up
Jenise
25%
Melissa
25 percent
Muzamil
950,000 - 712,500 = 237,500. 237,500 / 950,000 = .25 = 25%
Melissa
I've tried several times it won't let me post the breakdown of how you get 25%.
Melissa
Subtract one from the other to get the difference. Then take that difference and divided by 950000 and you will get .25 aka 25%
Melissa
Finally 👍
Melissa
one way is to set as ratio: 100%/950000 = x% / 712500, which yields that 712500 is 75% of the initial 950000. therefore, the decrease is 25%.
bruce
twenty five percent...
Jeorge
thanks melissa
Jeorge
950000-713500 *100 and then divide by 950000 = 25
Muzamil
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
6t+3
Melissa
6t +3
Bollywood
Tricia got a 6% raise on her weekly salary. The raise was $30 per week. What was her original salary? Iris Reply let us suppose her original salary is 'm'. so, according to the given condition, m*(6/100)=30 m= (30*100)/6 m= 500 hence, her original salary is$500.
Simply
28.50
Toi
thanks
Jeorge
How many pounds of nuts selling for $6 per pound and raisins selling for$3 per pound should Kurt combine to obtain 120 pounds of trail mix that cost him $5 per pound? Valeria Reply 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 code $20 per square foot. How many square feet of each tile should she use so that the overal cost of he backsplash will be$10 per square foot?
I need help with maths can someone help me plz.. is there a wats app group?
WY need
Fernando
How did you get $750? Laura Reply if y= 2x+sinx what is dy÷dx formon25 Reply does it teach you how to do algebra if you don't know how Kate Reply Liam borrowed a total of$35,000 to pay for college. He pays his parents 3% interest on the $8,000 he borrowed from them and pays the bank 6.8% on the rest. What average interest rate does he pay on the total$35,000? (Round your answer to the nearest tenth of a percent.)
exact definition of length by bilbao
the definition of length
literal meaning of length
francemichael
exact meaning of length
francemichael
exact meaning of length
francemichael
how many typos can we find...?
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
4x+7y=29,x+3y=11 substitute method of linear equation
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce