# 7.1 Greatest common factor and factor by grouping  (Page 2/5)

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$\begin{array}{cccc}\hfill 2\left(x+7\right)\hfill & & & \text{factors}\hfill \\ \hfill 2·x+2·7\hfill & & & \\ \hfill 2x+14\hfill & & & \text{product}\hfill \end{array}$

Now we will start with a product, like $2x+14$ , and end with its factors, $2\left(x+7\right)$ . To do this we apply the Distributive Property “in reverse.”

We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

## Distributive property

If $a,b,c$ are real numbers, then

$\begin{array}{ccccccc}a\left(b+c\right)=ab+ac\hfill & & & \text{and}\hfill & & & ab+ac=a\left(b+c\right)\hfill \end{array}$

The form on the left is used to multiply. The form on the right is used to factor.

So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!

## How to factor the greatest common factor from a polynomial

Factor: $4x+12$ .

## Solution

Factor: $6a+24$ .

$6\left(a+4\right)$

Factor: $2b+14$ .

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

## Factor the 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.

## Factor as a noun and a verb

We use “factor” as both a noun and a verb.

Factor: $5a+5$ .

## Solution

 Find the GCF of 5 a and 5. Rewrite each term as a product using the GCF. Use the Distributive Property "in reverse" to factor the GCF. Check by mulitplying the factors to get the orginal polynomial. $5\left(a+1\right)$ $5\cdot a+5\cdot 1$ $5a+5✓$

Factor: $14x+14$ .

$14\left(x+1\right)$

Factor: $12p+12$ .

$12\left(p+1\right)$

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Factor: $12x-60$ .

## Solution

 Find the GCF of 12 x and 60. Rewrite each term as a product using the GCF. Factor the GCF. Check by mulitplying the factors. $12\left(x-5\right)$ $12\cdot x-12\cdot 5$ $12x-60✓$

Factor: $18u-36$ .

$8\left(u-2\right)$

Factor: $30y-60$ .

$30\left(y-2\right)$

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

Factor: $4{y}^{2}+24y+28$ .

## Solution

We start by finding the GCF of all three terms.

 Find the GCF of $4{y}^{2}$ , $24y$ and 28. Rewrite each term as a product using the GCF. Factor the GCF. Check by mulitplying. $4\left({y}^{2}+6y+7\right)$ $4\cdot {y}^{2}+4\cdot 6y+4\cdot 7$ $4{y}^{2}+24y+28✓$

Factor: $5{x}^{2}-25x+15$ .

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

Factor: $3{y}^{2}-12y+27$ .

$3\left({y}^{2}-4y+9\right)$

Factor: $5{x}^{3}-25{x}^{2}$ .

## Solution

 Find the GCF of $5{x}^{3}$ and $25{x}^{2}.$ Rewrite each term. Factor the GCF. Check. $5{x}^{2}\left(x-5\right)$ $5{x}^{2}\cdot x-5{x}^{2}\cdot 5$ $5{x}^{3}-25{x}^{2}✓$

Factor: $2{x}^{3}+12{x}^{2}$ .

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

Factor: $6{y}^{3}-15{y}^{2}$ .

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

Factor: $21{x}^{3}-9{x}^{2}+15x$ .

## Solution

In a previous example we found the GCF of $21{x}^{3},9{x}^{2},15x$ to be $3x$ .

 Rewrite each term using the GCF, 3 x . Factor the GCF. Check. $3x\left(7{x}^{2}-3x+5\right)$ $3x\cdot 7{x}^{2}-3x\cdot 3x+3x\cdot 5$ $21{x}^{3}-9{x}^{2}+15x✓$

Factor: $20{x}^{3}-10{x}^{2}+14x$ .

$2x\left(10{x}^{2}-5x+7\right)$

Factor: $24{y}^{3}-12{y}^{2}-20y$ .

$4y\left(6{y}^{2}-3y-5\right)$

Factor: $8{m}^{3}-12{m}^{2}n+20m{n}^{2}$ .

## Solution

 Find the GCF of $8{m}^{3}$ , $12{m}^{2}n$ , $20m{n}^{2}$ . Rewrite each term. Factor the GCF. Check. $4m\left(2{m}^{2}-3mn+5{n}^{2}\right)$ $4m\cdot 2{m}^{2}-4m\cdot 3mn+4m\cdot 5{n}^{2}$ $8{m}^{3}-12{m}^{2}n+20m{n}^{2}✓$

Factor: $9x{y}^{2}+6{x}^{2}{y}^{2}+21{y}^{3}$ .

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

Factor: $3{p}^{3}-6{p}^{2}q+9p{q}^{3}$ .

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

When the leading coefficient is negative, we factor the negative out as part of the GCF.

Factor: $-8y-24$ .

## Solution

When the leading coefficient is negative, the GCF will be negative.

 Ignoring the signs of the terms, we first find the GCF of 8 y and 24 is 8. Since the expression −8 y − 24 has a negative leading coefficient, we use −8 as the GCF. Rewrite each term using the GCF. Factor the GCF. Check. $-8\left(y+3\right)$ $-8\cdot y+\left(-8\right)\cdot 3$ $-8y-24✓$

One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
rectangular field solutions
What is this?
Donna
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
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Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was$1,299. To the nearest tenth of a percent, what was the rate of discount?
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from$1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2% Orlando you could also just divide$725/$1299 and then subtract it from 1. then you get the same answer. Orlando p mulripied-5 and add 30 to it Tausif Reply p mulripied-5 and add30 Tausif p mulripied-5 and addto30 Tausif Can you explain further Monica Reply p mulripied-5 and add to 30 Tausif How do you find divisible numbers without a calculator? Jacob Reply TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13 BAINAMA When she graduates college, Linda will owe$43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan? Ariana Reply Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus? Kirisma Reply 66miles/hour snigdha How did you work it out? Esther s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr Orlando hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused. Alicia Reply what do you need help in? Felix subtracting a negative....is adding!! Heather look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number... Felix for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer... Felix —12 Niazmohammad Thanks Felix.l also get confused with signs. Esther Thank you for this Shatey ty Graham think about it like you lost$19 (-19), then found $7(+7). Totally you lost just$12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
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