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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 2 y 24 .
    If you missed this problem, review [link] .
  2. Factor 3 t 2 + 17 t + 10 .
    If you missed this problem, review [link] .
  3. Factor 36 p 2 60 p + 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.

This figure presents a general strategy for factoring polynomials. First, at the top, there is GCF, which is where factoring starts. Below this, there are three options, binomial, trinomial, and more than three terms. For binomial, there are the difference of two squares, the sum of squares, the sum of cubes, and the difference of cubes. For trinomials, there are two forms, x squared plus bx plus c and ax squared 2 plus b x plus c. There are also the sum and difference of two squares formulas as well as the “a c” method. Finally, for more than three terms, the method is grouping.

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 + b x + c ? Undo FOIL.
      Is it of the form a x 2 + b x + 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

Is there a GCF? Yes, 4 x 4 . 4 x 5 + 12 x 4 Factor out the GCF. 4 x 4 ( x + 3 ) In the parentheses, is it a binomial, a trinomial, or are there more than three terms? Binomial. Is it a sum? Yes. Of squares? Of cubes? No. Check. Is the expression factored completely? Yes. Multiply. 4 x 4 ( x + 3 ) 4 x 4 · x + 4 x 4 · 3 4 x 5 + 12 x 4

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Factor completely: 3 a 4 + 18 a 3 .

3 a 3 ( a + 6 )

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Factor completely: 45 b 6 + 27 b 5 .

9 b 5 ( 5 b + 3 )

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Factor completely: 12 x 2 11 x + 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.
.

This table has the heading of 12 x squared minus 11 x plus 2 and gives the possible factors. The first column is labeled possible factors and the second column is labeled product. Four rows have not an option in the product column. This is explained by the text, “if the trinomial has no common factors, then neither factor can contain a common factor”. The last factors, 3 x - 2 in parentheses and 4 x - 1 in parentheses, give the product of 12 x squared minus 11 x plus 2.

Check.

( 3 x 2 ) ( 4 x 1 )

12 x 2 3 x 8 x + 2

12 x 2 11 x + 2

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Factor completely: 10 a 2 17 a + 6 .

( 5 a 6 ) ( 2 a 1 )

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Factor completely: 8 x 2 18 x + 9 .

( 2 x 3 ) ( 4 x 3 )

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Factor completely: g 3 + 25 g .

Solution

Is there a GCF? Yes, g . g 3 + 25 g Factor out the GCF. g ( g 2 + 25 ) In the parentheses, is it a binomial, trinomial, or are there more than three terms? Binomial. Is it a sum ? Of squares? Yes. Sums of squares are prime. Check. Is the expression factored completely? Yes. Multiply. g ( g 2 + 25 ) g 3 + 25 g

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Factor completely: x 3 + 36 x .

x ( x 2 + 36 )

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Factor completely: 27 y 2 + 48 .

3 ( 9 y 2 + 16 )

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Factor completely: 12 y 2 75 .

Solution

Is there a GCF? Yes, 3. 12 y 2 75 Factor out the GCF. 3 ( 4 y 2 25 ) In the parentheses, is it a binomial, trinomial , or are there more than three terms? Binomial. Is it a sum? No. Is it a difference? Of squares or cubes? Yes, squares. 3 ( ( 2 y ) 2 ( 5 ) 2 ) Write as a product of conjugates. 3 ( 2 y 5 ) ( 2 y + 5 ) Check. Is the expression factored completely? Yes. Neither binomial is a difference of squares. Multiply. 3 ( 2 y 5 ) ( 2 y + 5 ) 3 ( 4 y 2 25 ) 12 y 2 75

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Questions & Answers

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.
Dojzae Reply
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
Xona Reply
5%
Michael
hey everyone how to do algebra
The Reply
Felecia answer 1.5 hours before he reaches her
Adriana Reply
I would like to solve the problem -6/2x
rachel Reply
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
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?
Stephanie Reply
Tickets for the community fair cost $12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold?
Alpha Reply
220
gayla
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
26+37=63+8=71
ziad
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
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
need help with math to do tsi test
Toocute
me too
Christian
have you tried the TSI practice test ***tsipracticetest.com
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

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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