<< Chapter < Page Chapter >> Page >
By the end of this section, you will be able to:
  • Factor perfect square trinomials
  • Factor differences of squares
  • Factor sums and differences of cubes
  • Choose method to factor a polynomial completely

Before you get started, take this readiness quiz.

  1. Simplify: ( 12 x ) 2 .
    If you missed this problem, review [link] .
  2. Multiply: ( m + 4 ) 2 .
    If you missed this problem, review [link] .
  3. Multiply: ( p 9 ) 2 .
    If you missed this problem, review [link] .
  4. Multiply: ( k + 3 ) ( k 3 ) .
    If you missed this problem, review [link] .

The strategy for factoring we developed in the last section will guide you as you factor most binomials, trinomials, and polynomials with more than three terms. We have seen that some binomials and trinomials result from special products—squaring binomials and multiplying conjugates. If you learn to recognize these kinds of polynomials, you can use the special products patterns to factor them much more quickly.

Factor perfect square trinomials

Some trinomials are perfect squares. They result from multiplying a binomial times itself. You can square a binomial by using FOIL, but using the Binomial Squares pattern you saw in a previous chapter saves you a step. Let’s review the Binomial Squares pattern by squaring a binomial using FOIL.

This image shows the FOIL procedure for multiplying (3x + 4) squared. The polynomial is written with two factors (3x + 4)(3x + 4). Then, the terms are 9 x squared + 12 x + 12 x + 16, demonstrating first, outer, inner, last. Finally, the product is written, 9 x squared + 24 x + 16.

The first term is the square of the first term of the binomial and the last term is the square of the last. The middle term is twice the product of the two terms of the binomial.

( 3 x ) 2 + 2 ( 3 x · 4 ) + 4 2 9 x 2 + 24 x + 16

The trinomial 9 x 2 + 24 +16 is called a perfect square trinomial. It is the square of the binomial 3 x +4.

We’ll repeat the Binomial Squares Pattern here to use as a reference in factoring.

Binomial squares pattern

If a and b are real numbers,

( a + b ) 2 = a 2 + 2 a b + b 2 ( a b ) 2 = a 2 2 a b + b 2

When you square a binomial, the product is a perfect square trinomial. In this chapter, you are learning to factor—now, you will start with a perfect square trinomial and factor it into its prime factors.

You could factor this trinomial using the methods described in the last section, since it is of the form ax 2 + bx + c . But if you recognize that the first and last terms are squares and the trinomial fits the perfect square trinomials pattern    , you will save yourself a lot of work.

Here is the pattern—the reverse of the binomial squares pattern.

Perfect square trinomials pattern

If a and b are real numbers,

a 2 + 2 a b + b 2 = ( a + b ) 2 a 2 2 a b + b 2 = ( a b ) 2

To make use of this pattern, you have to recognize that a given trinomial fits it. Check first to see if the leading coefficient is a perfect square, a 2 . Next check that the last term is a perfect square, b 2 . Then check the middle term—is it twice the product, 2 ab ? If everything checks, you can easily write the factors.

How to factor perfect square trinomials

Factor: 9 x 2 + 12 x + 4 .

Solution

This table gives the steps for factoring 9 x squared +12 x +4. The first step is recognizing the perfect square pattern “a” squared + 2 a b + b squared. This includes, is the first term a perfect square and is the last term a perfect square. The first term can be written as (3 x) squared and the last term can be written as 2 squared. Also, in the first step, the middle term has to be twice “a” times b. This is verified by 2 times 3 x times 2 being 12 x. The second step is writing the square of the binomial. The polynomial is written as (3 x) squared + 2 times 3 x times 2 + 2 squared. This is factored as (3 x + 2) squared. The last step is to check with multiplication.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Factor: 4 x 2 + 12 x + 9 .

( 2 x + 3 ) 2

Got questions? Get instant answers now!

Factor: 9 y 2 + 24 y + 16 .

( 3 y + 4 ) 2

Got questions? Get instant answers now!

The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern a 2 2 a b + b 2 , which factors to ( a b ) 2 .

The steps are summarized here.

Factor perfect square trinomials.

Step 1. Does the trinomial fit the pattern? a 2 + 2 a b + b 2 a 2 2 a b + b 2 Is the first term a perfect square? ( a ) 2 ( a ) 2 Write it as a square. Is the last term a perfect square? ( a ) 2 ( b ) 2 ( a ) 2 ( b ) 2 Write it as a square. Check the middle term. Is it 2 a b ? ( a ) 2 2 · a · b ( b ) 2 ( a ) 2 2 · a · b ( b ) 2 Step 2. Write the square of the binomial. ( a + b ) 2 ( a b ) 2 Step 3. Check by multiplying.

Questions & Answers

4x+7y=29,x+3y=11 substitute method of linear equation
Srinu Reply
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
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
Andrew Reply
divide 3x⁴-4x³-3x-1 by x-3
Ritik Reply
how to multiply the monomial
Ceny Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
Seera Reply
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Juned Reply
Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes
ashley Reply
app is wrong how can 350 be divisible by 3.
Raheem Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna Reply
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
Lorris Reply
I'm getting "math processing error" on math problems. Anyone know why?
Ray Reply
Can you all help me I don't get any of this
Jade Reply
4^×=9
Alberto Reply
Did anyone else have trouble getting in quiz link for linear inequalities?
Sireka Reply
operation of trinomial
Justin Reply
Practice Key Terms 3

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