<< Chapter < Page Chapter >> Page >
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

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Factor completely: 3 a 4 + 18 a 3 .

3 a 3 ( a + 6 )

Got questions? Get instant answers now!

Factor completely: 45 b 6 + 27 b 5 .

9 b 5 ( 5 b + 3 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Factor completely: 10 a 2 17 a + 6 .

( 5 a 6 ) ( 2 a 1 )

Got questions? Get instant answers now!

Factor completely: 8 x 2 18 x + 9 .

( 2 x 3 ) ( 4 x 3 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Factor completely: x 3 + 36 x .

x ( x 2 + 36 )

Got questions? Get instant answers now!

Factor completely: 27 y 2 + 48 .

3 ( 9 y 2 + 16 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

rectangular field solutions
Navin Reply
What is this?
Donna
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
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?
Manhwa Reply
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
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
where's the answers?
Ed Reply
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?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply
What does 3 times your weight right now
Cherokee Reply
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

Get the best Elementary algebra course in your pocket!





Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask