Chapter 5: Factoring
5.1: Factoring by
GCF and Grouping
5.2: Factoring
Trinomials Form 
5.3: Factoring
Trinomials Form 
5.4: Factoring
Difference of Two Squares
and Factor Perfect Square Trinomials
5.5: Factoring
Using Multiple Methods
5.6: Solve Quadratic
Equations by Factoring
5.7: Simplify
Rational Expressions
5.8: Factoring
Applications
5.1: Factoring by GCF
and Grouping
What is factoring?
Find the prime factors of:
6 27 250
The greatest common factor (GCF)- the largest common factor of the integers.
To Find the GCF:
1. Find the prime factorization of the terms
2. Find the common factors in each of the terms and the most of each in all terms
3. Multiply the most common factors in all of the terms.
Example: Find the GCF of: 36 and 90
The common factors are: 2, 3
The most 2’s in common is: 2
The most 3’s in common is: 
The GCF is: 
Practice Example: Find the GCF of: 24 and 70
Practice Example:
Find the GCF of: 45 60 75
Example: Find the GCF of: 
and 
The common factors are: 3,x, y
The most 3’s in common is: 
The most x’s in common is: 
The most y’s in common is: 
The GCF is: 
Practice Example: Find the GCF of: 
and 
Practice Example:
Find the GCF of: 
Simplify: 
Now work backwards to get the original back
Factor using Greatest Common Factor
Example: Factor- 
If we break down the polynomial into prime factors we would have:
What are the most factors you can pull out of all of the terms?
Then what remains once that is pulled out?
Divide by what you are factoring out to see what remains.
=
The factored form
is: 
You can check by simplifying.
Practice Examples:
Factor:
Factor:
Factor:
Factor using Grouping
Factor:
Example: Factor- 
Method 1 will not work because you can not pull out of all of the terms.
Since there are 4 terms, we can group 2 in common and another 2 in common.
and 
Now, we can pull out from each group:
and 
If this is factorable, then what is the parenthesis will match.
Pull out what they both have in common and see what remains.
, this is the factored form.
You can check by simplifying.
Practice Example:
Factor:
Practice Examples:
Factor:
Factor:
Factor:
What is factoring?
______________________________________________________________________________
______________________________________________________________________________
5.1: Factoring by GCF and Grouping Practice Problems
1. Find the GCF of: 2. Find the GCF of:
54, 72, 90 
, 
3. Find the GCF of: 4. Find the GCF of:
, 
, 
, 
, 
5. Factor: 6. Factor:
7. Factor: 8. Factor:
5.1: Factoring by GCF and Grouping Practice Problems Continue
9. Factor: 
10.
Factor: 
11. Factor: 
12. Factor: 
13. Factor: 
14. Factor: 
15. Factor: 
16. Factor: 
5.2: Factoring
Trinomials Form
How do you factor
trinomials in the form ?
Simplify: 
Solution:
Notice to get the first term, it is the multiplication of the F part of FOIL
Notice to get the inside term, it is the addition of the O + I of FOIL
Notice to get the outside term, it is the multiplication of L of FOIL.
Simplify: 
Example: Factor- 
Pulling out the GCF or Grouping will not work because you can not pull out of all of the terms and there is not an even number of terms to group.
Our goal is to factor into two factors that are being
multiplied: 
To get the
, the factors must be 
in the front (F
of FOIL).
To factor this polynomial where the leading coefficient is 1:
We need to two numbers that will multiply to the last term of: + 12 (L of FOIL)
and add to the middle term of: + 7 (O + I of FOIL)
We look at the factors of +12 and find: 
, 
, 
, 
, 
Which pair will multiply to +12 and add to +7?
The numbers are: +3 and +4. 
****Make sure your signs are correct***
The factored form is:
You can check by simplifying.
Know Your Signs
Multiply: +
Then the signs are:
+ + OR - -
 |
 |
||
Add: + Add: -
+ + - -
________________________________________________________________________
Multiply: -
Then the signs are:
+ -
Practice Examples:
Factor: Factor:
Factor: Factor:
Factor: Factor:
How do you factor
trinomials in the form ?
______________________________________________________________________________
______________________________________________________________________________
5.2: Factoring Trinomials
Form Practice
Examples
1. Factor: 
2.
Factor:
3. Factor: 
4.
Factor: 
5. Factor: 
6.
Factor: 
7. Factor: 
8.
Factor: 
5.2: Factoring Trinomials
Form 
Practice
Examples Continue
9. Factor: 
10.
Factor: 
11. Factor: 
12.
Factor: 
13. Factor: 
14.
Factor: 
15. Factor: 
16.
Factor: 
5.3: Factoring Trinomials
Form
How do you factor trinomials
in the form ?
Example: Factor- 
Method 1 or Method 2 or Method 3 will not work because:
You can not pull out of all of the terms (Method 1 Pull out GCF),
There is not an even number of terms to group (Method 2 Grouping), and
Since the leading coefficient is not 1, (Method 3 Trinomial a = 1)
You still need two numbers that multiply to the last term, but they will NOT add to the middle term coefficient because the leading coefficient is not 1.
There are two ways to factor this polynomial: Trial and Error and Change to grouping
First Method: Trial and Error
We know that when we factor, the factored form will be:
Step 1: We will first list the possible factors:
We must get the leading term of 
The possibilities are: 
Step 2: We know that it must multiply to the last term of + 5:
The possibilities with signs are: 
**Signs
are VERY important**
Since the middle term is negative, and it must add to the
middle term, then it must be:
Step 3: Then try different combinations and simply
each one to try to get:
Ex. 
, this is the factored form. You
can check by simplifying.
Second Method: Change to grouping
Example: Factor-
In this method we want the trinomial to become a polynomial with four terms so we can perform grouping. There is a specific way to change the trinomial into a polynomial with four terms.
Step 1: Find the middle factors
Multiply the leading coefficient by the last constant term.
, we want to find factors that
multiply to get this number (including the sign)
The factors must also add to the middle term coefficient, in this case -23.
The factors that multiply to +60 and add to -23 are: -20 and -3
Step 2: Rewrite the polynomial
We found the two coefficients to replace the middle term to form a polynomial with 4 terms.
The trinomial will now become:
It does not matter which number goes first, but since we are going to group, you want to place factors next to terms they have something in common with.
For example, place the “-3x” next to the “
” because they
have a “3x” in common and place the “-20x” next to the “5” because they have a
“5” in common.
Step 3: Use Grouping
, this is the factored form. You
can check by simplifying.
Practice Example:
Factor: Factor:
Factor: Factor:
Factor:
Factor:
How do you factor
trinomials in the form ?
______________________________________________________________________________
______________________________________________________________________________
5.3: Factoring Trinomials
Form Practice
Examples
1. Factor: 
2.
Factor: 
3. Factor: 
4.
Factor: 
5. Factor: 
6.
Factor: 
5.3: Factoring Trinomials
Form 
Practice
Examples Continue
7. Factor: 
8.
Factor: 
9. Factor: 
10.
Factor: 
11. Factor: 
12.
Factor: 
5.4: Factoring Difference of Two Squares
and Factor Perfect Square Trinomials
How do you factor difference of two squares?
Binomial (+/-) Special Case (Difference of Two Squares)
Example: Factor- 
Method 1 or Method 2 or Method 3 or Method 4 will not work because you can not pull out of all of the terms, there is not an even number of terms to group, and it is not a trinomial.
This is a Special Case because:
1) There are only 2 terms
2) There is a (-) in the middle
3) You can take the square root of both terms
If all the criteria above represented, then you have difference of squares.
To factor this polynomial, you take the square root of both terms:
2x and 3
Since there is no middle term, it must have canceled each other out.
Therefore, you make one factor with (+) and another factor with (-).
, this is the factored form.
You can check by simplifying.
Practice Example:
Factor: Factor:
Factor: Factor:
Factor: Factor:
How do you factor difference of two squares?
______________________________________________________________________________
______________________________________________________________________________
Factoring Perfect Square Trinomial
Example: Factor- 
is a perfect square trinomial
because:
The first term: is the square of x 
The last term: 25 is the square of
5: 
The second term: 10x is twice the product of x and 5: 2(5)(x) = 10x
A trinomial is a perfect square trinomial if:
The factored form of: is: 
________________________________________________________________________
Example: Factor- 
is a perfect square trinomial
because:
The first term: 
is the square of 3x 
The last term: 49 is the square of
7: 
The second term: -42x is - twice the product of 3x and 7: -2(3x)(7) = -42x
A trinomial is a perfect square trinomial if:
The factored form of: is: 
Practice Example:
Factor: Factor:
Factor: Factor:
5.4: Factoring Difference of Two Squares and Factor Perfect Square Trinomials Practice Problems
Factor the following:
1. 
2. 
3. 
4. 
5.
6.
7. 
8. 
9.
5.4: Factoring Difference of Two Squares and Factor Perfect Square Trinomials Practice Problems Continue
Factor the following:
10. 
11. 
12.
13. 
14. 
15.
16. 
17. 
18.
5.5: Factoring Using Multiple Methods
If sometimes you need to factor using multiple methods,
what process should you follow?
Steps for Factoring Polynomial
Check each method in this order:
1) Pull out the GCF
Ex.
2) Grouping (Hint: 4 Terms or terms that can be paired)
Ex.
4) Trinomial where leading coefficient is 1
Ex.
5) Trinomial where the leading coefficient is not 1
Ex.
6) Difference of Squares
Ex.
If sometimes you need to factor using multiple methods,
what process should you follow?
__________________________________________________________________
__________________________________________________________________
Practice Examples:
Factor: Factor:
Factor: Factor:
5.5: Factoring Using Multiple Methods Practice Problems
1. Factor: 2. Factor:
3. Factor: 4. Factor:
5. Factor: 6. Factor:
5.6: Solve Quadratic Equations by
Factoring
How do you solve quadratic equations by factoring?
Example: Solve: 
by Factoring
In this method, MAKE SURE that the quadratic equation is equal to 0
Step 1: Factor the quadratic equation
Step 2: Set each factor equal to 0
Step 3: Solve each equation
Step 4: Check your answers
Answers are x = 0.5 and x = -3
Substitute each answer individually into the original equation:
Example: Solve the quadratic equation: 
Step 1: Set the quadratic equation equal to zero
Step 2: Factor the quadratic equation
Step 3: Set each factor equal to zero
Step 4: Solve each factor
Step 5: Check the solutions
Algebraically
Practice Example: Solve: 
by Factoring
EXAMPLE: Solve the quadratic equation: 
Step 1: Set the quadratic equation equal to zero
Step 2: Factor the quadratic equation
Step 3: Set each factor equal to zero
Step 4: Solve each factor
Step 5: Check the solutions
How do you solve quadratic equations by factoring?
______________________________________________________________________________
______________________________________________________________________________
5.6: Solve Quadratic Equations by Factoring Practice Problems
Solve:
1. 
2.
3. 
4.
5. 
6.
5.6: Solve Quadratic Equations by Factoring Practice Problems Continue
Solve:
7. 
8.
9. 
10.
11. 
12.
5.7: Simplify Rational Expressions
How do you simplify rational expressions?
Example: Simplify- 
Solution:
Step 1: Factor the numerator and denominator
Step 2: Simplify
Step 3: Final Answer
Practice Examples:
Simplify- 
Simplify- 
Simplify- 
Simplify- 
Example: Simplify- 
Solution:
Step 1: Factor the numerator and denominator
Step 2: Simplify
Step 3: Final Answer
Practice Examples:
Simplify- 
Simplify- 
Practice Examples:
Simplify- 
Simplify-
Simplify- 
Simplify- 
How do you simplify rational expressions?
______________________________________________________________________________
______________________________________________________________________________
5.7: Simplify Rational Expressions Practice Problems
Simplify:
1. 
2.
3. 
4.
5. 
6.
7. 
8.
5.7: Simplify Rational Expressions Practice Problems Continue
Simplify:
9. 
10.
11. 
12.
13. 
14.
5.8: Factoring Applications
What are some examples where factoring is used in life?
Word Problem:
Find the area of a rectangle given that the length of the rectangle is 5 more than twice the width.
Solution:
|
x
To find the area
of a rectangle use the formula for Area of a rectangle: Area = Length times
Width or 
In this case, the width is “x” and the length is “2x+5” so the area is:
Factoring Application 1: Factoring can be used to find unknowns.
Example:
Find the length
and width of a rectangle whose area is: 
(In this case, the length is the longer side and the width is the shorter side.)
Once the width and length algebraic expressions are found, find the value of the width and length when x = 5 ft. Then verify the area by substituting x = 5 ft into the area expression.
|
?
Solution:
To find the solution factor the area to find the length and width:
The width is “x + 2” feet and the length “2x + 1” feet.
|
x + 2
If x = 5 ft, then:
Practice Example:
Find the length
and width of a rectangle garden whose area is: 
(In this case the length is the longer side and the width is the shorter side.)
Once the width and length are found, find the value of the width and length when x = 10 ft. Then verify the area by substituting x = 10 ft into the area.
|
?=
Practice Example:
Find the length
and width of a square play area whose area is: 
(In this case the length is the longer side and the width is the shorter side.)
Once the width and length are found, find the value of the width and length when x = 6 ft. Then verify the area by substituting x = 6 ft into the area.
|
?=
Factoring Application 2: Solving real life equations
The equation of a person who dives off a 48 ft. cliff into a river with an initial velocity of 32 ft/sec is:
, where
“t” is time in seconds and “h” is height in feet.
When will the diver hit the water (when height = 0)?
Solution:
To find the solution:
1) Substitute 0 in for the height (h)
2) Factor the trinomial by first factoring out the -16:
3) Continue factoring by factoring the trinomial:
4) Set each term with a variable equal to the right side of 0 and solve for t:
5) The solutions are 3 seconds and -1 seconds. Since -1 seconds is not realistic, we do not include that solution. The diver will hit the water is 3 seconds.
Practice Example: 
The equation of a person who dives off a 64 ft. cliff into a river with an initial velocity of 0 ft/sec is:
, where
“t” is time in seconds and “h” is height in feet.
When will the diver hit the water (when height = 0)?
Practice Example: 
The equation of
the path of a ball thrown in the air at a speed of 96 ft/sec is: 
, where “t” is time (seconds) and “h” is
height in feet.
When will the ball hit the ground (when height = 0)?
What are some examples where factoring is used in life?
____________________________________________________
____________________________________________________
5.8: Factoring Applications Practice Problems
Solve:
1. Find the
length and width of a rectangular room whose area is: 
Draw a picture of the situation.
(In this case the length is the longer side and the width is the shorter side.)
Once the width and length are found, find the value of the width and length when x = 10 ft. Then verify the area by substituting x = 10 ft into the area.
2. Find the
length and width of a square room whose area is: 
Draw a picture of the situation.
(In this case the length is the longer side and the width is the shorter side.)
Once the width and length are found, find the value of the width and length when x = 4 ft. Then verify the area by substituting x = 4 ft into the area.
5.8: Factoring Applications Practice Problems Continue
Solve:
3. The equation of a person who dives off a 96 ft. cliff into a river with an initial velocity of 16 ft/sec is:
, where
“t” is time in seconds and “h” is height in feet.
When will the diver hit the water (when height = 0)?
4. The equation of
the path of a ball thrown in the air at a speed of 16 ft/sec is: 
, where “t” is time (seconds) and “h” is
height in feet.
When will the ball hit the ground (when height = 0)?
MAT0024C Beginning Algebra Name: ________________
Test 3 (Chapter 5) REVIEW
Questions from Chapter 5
Factor:
1. 
2.
3. 
4. 
5. 
6.
7. 
8.
9. 
10.
11. 
12.
13. 
14.
Solve:
15. 
= 0 16.
=
0
Solve:
17. 
= 0 18.
19. Simplify: 
20.
Simplify: 
21. Two machines can complete 8 tasks every 3 days. Let t represent the number of tasks these machines can complete in a 30-day month. Write a proportion to show this example.
22. Simplify: 
23.
Convert to scientific notation: 0.000023
24. Solve: 
25.
Simplify: 
26. Simplify: 
27.
Solve for t: 
28. Find the y –intercept for: 
29. Graph: 
30. The length of a rectangular pool is 6 less than twice the width. The perimeter of the pool is 78 feet. Find the length and width of the pool. Label each distance correctly.