# 7.5 General strategy for factoring polynomials

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By the end of this section, you will be able to:
• Recognize and use the appropriate method to factor a polynomial completely

Before you get started, take this readiness quiz.

1. Factor ${y}^{2}-2y-24$ .
If you missed this problem, review [link] .
2. Factor $3{t}^{2}+17t+10$ .
If you missed this problem, review [link] .
3. Factor $36{p}^{2}-60p+25$ .
If you missed this problem, review [link] .
4. Factor $5{x}^{2}-80$ .
If you missed this problem, review [link] .

## Recognize and use the appropriate method to factor a polynomial completely

You have now become acquainted with all the methods of factoring that you will need in this course. (In your next algebra course, more methods will be added to your repertoire.) The figure below summarizes all the factoring methods we have covered. [link] outlines a strategy you should use when factoring polynomials.

## 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?

Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

Factor completely: $4{x}^{5}+12{x}^{4}$ .

## Solution

$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes,}\phantom{\rule{0.2em}{0ex}}4{x}^{4}.\hfill & & & \hfill 4{x}^{5}+12{x}^{4}\hfill \\ & & & \text{Factor out the GCF.}\hfill & & & \hfill 4{x}^{4}\left(x+3\right)\hfill \\ \text{In the parentheses, is it a binomial, a}\hfill & & & & & & \\ \text{trinomial, or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Is it a sum?}\hfill & & & & & & \text{Yes.}\hfill \\ \phantom{\rule{1em}{0ex}}\text{Of squares? Of cubes?}\hfill & & & & & & \text{No.}\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely?}\hfill & & & & & & \text{Yes.}\hfill \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \\ \\ \phantom{\rule{2.5em}{0ex}}4{x}^{4}\left(x+3\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}4{x}^{4}·x+4{x}^{4}·3\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}4{x}^{5}+12{x}^{4}\phantom{\rule{0.2em}{0ex}}✓\hfill & & & & & & \end{array}$

Factor completely: $3{a}^{4}+18{a}^{3}$ .

$3{a}^{3}\left(a+6\right)$

Factor completely: $45{b}^{6}+27{b}^{5}$ .

$9{b}^{5}\left(5b+3\right)$

Factor completely: $12{x}^{2}-11x+2$ .

## Solution Is there a GCF? No. Is it a binomial, trinomial, or are there more than three terms? Trinomial. Are a and c perfect squares? No, a = 12, not a perfect square. Use trial and error or the “ac” method. We will use trial and error here.  Check.

$\phantom{\rule{2.5em}{0ex}}\left(3x-2\right)\left(4x-1\right)$

$\phantom{\rule{2.5em}{0ex}}12{x}^{2}-3x-8x+2$

$\phantom{\rule{2.5em}{0ex}}12{x}^{2}-11x+2\phantom{\rule{0.2em}{0ex}}✓$

Factor completely: $10{a}^{2}-17a+6$ .

$\left(5a-6\right)\left(2a-1\right)$

Factor completely: $8{x}^{2}-18x+9$ .

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

Factor completely: ${g}^{3}+25g$ .

## Solution

$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes,}\phantom{\rule{0.2em}{0ex}}g.\hfill & & & \hfill {g}^{3}+25g\hfill \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill g\left({g}^{2}+25\right)\hfill \\ \text{In the parentheses, is it a binomial, trinomial,}\hfill & & & & & & \\ \text{or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Is it a sum ? Of squares?}\hfill & & & \text{Yes.}\hfill & & & \text{Sums of squares are prime.}\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely?}\hfill & & & \text{Yes.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}g\left({g}^{2}+25\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}{g}^{3}+25g\phantom{\rule{0.2em}{0ex}}✓\hfill & & & & & & \end{array}$

Factor completely: ${x}^{3}+36x$ .

$x\left({x}^{2}+36\right)$

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

$3\left(9{y}^{2}+16\right)$

Factor completely: $12{y}^{2}-75$ .

## Solution

$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 3.}\hfill & & & \hfill 12{y}^{2}-75\hfill \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill 3\left(4{y}^{2}-25\right)\hfill \\ \text{In the parentheses, is it a binomial, trinomial},\hfill & & & & & & \\ \text{or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & \\ \text{Is it a sum?}\hfill & & & \text{No.}\hfill & & & \\ \text{Is it a difference? Of squares or cubes?}\hfill & & & \text{Yes, squares.}\hfill & & & \hfill 3\left({\left(2y\right)}^{2}-{\left(5\right)}^{2}\right)\hfill \\ \text{Write as a product of conjugates.}\hfill & & & & & & \hfill 3\left(2y-5\right)\left(2y+5\right)\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely?}\hfill & & & \text{Yes.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Neither binomial is a difference of}\hfill & & & & & & \\ \phantom{\rule{1em}{0ex}}\text{squares.}\hfill & & & & & & \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}3\left(2y-5\right)\left(2y+5\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}3\left(4{y}^{2}-25\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}12{y}^{2}-75\phantom{\rule{0.2em}{0ex}}✓\hfill & & & & & & \end{array}$

#### Questions & Answers

Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
Amara currently sells televisions for company A at a salary of $17,000 plus a$100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a$20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
Wenda Reply
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned$1600 more than Jim earned. How much did they earned
Arleathia Reply
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
Sunnyshay Reply
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
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?
Gagan Reply
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
1,75 hrs is correct Mike
Emund
How did you come up with the answer?
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Marie Reply
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have? Kendra Reply Tickets to a Broadway show cost$35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were$47,150. How many adult and how many child tickets were sold?
dana Reply
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year? How do I do this
Tanesia Reply
how to square
Fiona Reply
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
Sabee Reply

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