1. Factoring Expressions

2. Definitions and Key Ideas
Factoring is the process of writing an expression as a multiplication problem.
Factoring “undoes” multiplication
We can write numbers in terms of factors:
What are the factors of 12?
To factor 12, we would write it as a product of its factors:
What is special about prime factors?
These are special factors of 12 known as prime factors.

3. Algebraic Expressions
There are several different methods used to factor algebraic expressions, or expressions that contain variables.
All of our factoring methods are built around the idea of area of a rectangle.
Because factoring is the process of “undoing” multiplication, let’s review our multiplication techniques.
AreaRectangle = (base)(height)

4. Multiplication Review
Example 1:
There are 2 methods used to multiply these expressions.
Distributive Property
Area and Rectangles
Multiply the inside terms by the outside value. 2x + 3 4 8x 12
A = bh
The total area of the large rectangle is 8x + 12.

5. Factoring We need to start with our solution:
How can we work backwards from this process?
x x
Let’s look at the dimensions of the rectangle we made.
Think about the algebra tiles– can we build a rectangle? 4
Simplify the base! 2x
Area = (base)(height)
8x + 12 = (2x + 3)(4)
** Don’t forget that using generic rectangles can save time!

6. Factoring Example 2: Using Generic Rectangles x + 2
When asked to factor, you are given the area and are asked to find the dimensions. 4x
1) Find a height that works for both small rectangles.
When finding the dimensions you want to find the largest height possible. This height is the GREATEST COMMON FACTOR!
2) Use that height and the areas to find the base.
The factored form: 4x(x+2)

7. Practice
Use the concept of area to find the factored form of each of the following expressions.
Solutions:
Once you have completed all the problems, click the mouse again for the solutions!

8. Multiplication Review
Example 2
Again we are going to use the concept of area to multiply our 2 factors. x x + 5
Don’t forget to combine like terms!
Use each of the factors for either the base or the height.
Multiply base and height to find the area.
** Because it is a multiplication problem, it does not matter which factor is the base and which is the height.

9. Factoring Method 1: Generic Rectangle
Remember that factoring is working backwards to write a multiplication problem.
Factoring
When given an expression with 3 terms, there are 2 methods of factoring that can be used.
Method 1: Generic Rectangle
Once all of the areas are labeled, we have to find the dimensions. 5x
Label each of the individual rectangles, then find the dimensions of the larger rectangle. 4x
We use 4 & 5 because they also multiply to equal 20! 4x 5x

10. Factoring A M Method 2: Diamond Problems
Note in the previous example that in order to fill in the areas for the smaller rectangles, we needed to find 2 numbers that added up to 9 and multiplied to 20.
Anytime you have an expression with 3 terms and there is not a coefficient with we can use a diamond problem to find our factors. 20
This is the same process we used to complete diamond problems. 4 5 9

11. Practice
Use either generic rectangles or diamonds to factor each of the following.
Once you have factored all of these, click the mouse again to check your answers!
Remember, the order of the factors does not matter!

12. Special Cases
There are 2 special cases for factoring: Difference of Squares and Perfect Square Trinomial.
Difference of Squares
Example:
Difference of squares is used when you have 2 terms separated by subtraction.
1) Rewrite the expression with 3 terms.
2) Use a diamond problem to factor.
-16
3) Write final factors. -4 4

13. Special Cases Perfect Square Square Trinomials
A perfect square trinomial looks just like a diamond problem. The difference is in how we write the answer. 36
1) Factor using a diamond. 6 6 12
2) What do you notice about your solution? Can we write it a simpler way?

14. Practice
In each of the problems below, first decide whether you have a difference of squares or a perfect square trinomial, then factor.
Solve each of the problems. After you finish, click the mouse to check your answers.

15. Now that we have talked about several methods of factoring, let’s put them together!!
Factoring Completely
Factoring completely is a combination of generic rectangles and diamond problems to present our answer in simplest form!
1) Set up a generic rectangle to factor the GCF– find the largest height!
2) Find the base of the rectangle.
3) Try to write the base with 3 terms to factor. -4
4) Now, we can use a diamond to factor the base. 2 -2
5) Write your final expression.

16. An Exception– ax2 6 6 1 7 Not all expressions have a GCF.
Note that the terms in the example do not have anything in common, but there is still a coefficient (3) in front of x2.
Not all expressions have a GCF.
We need to use a modified diamond.
1. Multiply the 1st and 3rd terms.
2. Set up a diamond problem with the product in the top. 6
3. Rewrite the expression with 4 terms. 7
4. Use these terms to fill in the area of a generic rectangle.
5. Find the dimensions and write the factors.

17. Practice
Factor each of the polynomials completely. Don’t forget to start with the GCF (largest height) first!
After you factor each problem, click the mouse again to check your answers.

18. Remember: In order to factor, start with a GCF (largest height) first
Remember: In order to factor, start with a GCF (largest height) first. Then, look at the number of terms in the base to decide where to go next!
Review
3 terms:
2 Terms: Difference of Squares
Modified Diamond
Diamond Problem