9.7: Factoring Polynomials Completely
We say that a polynomial is factored completely when we factor as much as we can and we are unable to factor any more. Here are some suggestions that you should follow to make sure that you factor completely.
Example 1: Factor the following polynomials completely.
(a)
(b)
Solution:
(a) Look for the common monomial factor.
(b) Recognize this as a perfect square and factor as
Factoring Common Binomials
The first step in the factoring process is often factoring the common monomials from a polynomial. Sometimes polynomials have common terms that are binomials. For example, consider the following expression.
You can see that the term
This expression is now completely factored. Let’s look at some examples.
Example 2: Factor
Solution:
When we factor the common binomial, we get
Factoring by Grouping
It may be possible to factor a polynomial containing four or more terms by factoring common monomials from groups of terms. This method is called factoring by grouping. The following example illustrates how this process works.
Example 3: Factor
Solution: There isn't a common factor for all four terms in this example. However, there is a factor of 2 that is common to the first two terms and there is a factor of
Now we notice that the binomial
Our polynomial is now factored completely.
We know how to factor Quadratic Trinomials
 We find the product
ac .  We look for two numbers that multiply to give
ac and add to giveb .  We rewrite the middle term using the two numbers we just found.
 We factor the expression by grouping.
Let’s apply this method to the following examples.
Example 4: Factor
Solution: Follow the steps outlined above.
The number 12 can be written as a product of two numbers in any of these ways:
Rewrite the middle term as:
Factor an
Now factor the common binomial
Our answer is
In this example, all the coefficients are positive. What happens if the
Example 5: Factor
Solution:
The number 24 can be written as a product of two numbers in any of these ways.
Rewrite the middle term as
Factor by grouping. Factor a
Now factor the common binomial
Our answer is
Solving RealWorld Problems Using Polynomial Equations
Now that we know most of the factoring strategies for quadratic polynomials, we can see how these methods apply to solving realworld problems.
Example 6: The product of two positive numbers is 60. Find the two numbers if one of the numbers is 4 more than the other.
Solution:
Write the polynomial in standard form.
Factor:
The expression factors as
Solve:
Since we are looking for positive numbers, the answer must be positive.
Check:
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK12 Basic Algebra: Factor by Grouping and Factoring Completely (13:57)
Factor completely.

2x2+16x+30 
12c2−75 
−x3+17x2−70x 
6x2−600 
−5t2−20t−20 
6x2+18x−24 
−n2+10n−21 
2a2−14a−16 
2x2−512 
12x3+12x2+3x
Factor by grouping.

6x2−9x+10x−15 
5x2−35x+x−7 
9x2−9x−x+1 
4x2+32x−5x−40 
12x3−14x2+42x−49 
4x2+25x−21 
24b3+32b2−3b−4 
2m3+3m2+4m+6 
6x2+7x+1 
4x2+8x−5 
5a3−5a2+7a−7 
3x2+16x+21 
4xy+32x+20y+160 
10ab+40a+6b+24 
9mn+12m+3n+4 
4jk−8j2+5k−10j 
24ab+64a−21b−56
Solve the following application problems.
 One leg of a right triangle is seven feet longer than the other leg. The hypotenuse is 13 feet. Find the dimensions of the right triangle.
 A rectangle has sides of
x+2 andx−1 . What value ofx gives an area of 108?  The product of two positive numbers is 120. Find the two numbers if one numbers is seven more than the other.
 Framing Warehouse offers a pictureframing service. The cost for framing a picture is made up of two parts. The cost of glass is $1 per square foot. The cost of the frame is $2 per linear foot. If the frame is a square, what size picture can you get framed for $20.00?
Mixed Review
 The area of a square varies directly with its side length.
 Write the general variation equation to model this sentence.
 If the area is 16 square feet when the side length is 4 feet, find the area when
s=1.5 feet .
 The surface area is the total amount of surface of a threedimensional figure. The formula for the surface area of a cylinder is
SA=2πr2+2πrh , wherer=radius andh=height of the cylinder . Determine the surface area of a soup can with a radius of 2 inches and a height of 5.5 inches.  Factor
25g2−36 . Solve this polynomial when it equals zero.  What is the greatest common factor of
343r3t,21t4 , and63rt5 ?  Discounts to the hockey game are given to groups with more than 12 people.
 Graph this solution on a number line
 What is the domain of this situation?
 Will a church group with 12 members receive a discount?
Image Attributions
Description
Tags:
Subjects:
Date Created:
Feb 22, 2012Last Modified:
Dec 11, 2014If you would like to associate files with this None, please make a copy first.