Factoring ExpressionsReview

Definitions and Key IdeasFactoring is the process of writing an expression as a multiplication problem.We can write numbers in terms of factors:

What are the factors of 12?12 6, 4, 3, 2, 1, Answer

To factor 12, we would write it as a product of its factors: 322

These are special factors of 12 known as prime factors.

What is special about prime factors?

Factoring “undoes” multiplication

Algebraic ExpressionsThere 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.

AreaRectangle = (base)(height)

Because factoring is the process of “undoing” multiplication, let’s review our multiplication techniques.

Multiplication ReviewExample 1: )32(4 x

There are 2 methods used to multiply these expressions.

Distributive PropertyMultiply the inside terms by the outside value.

)32(4 x3424 x

128 x

Area and Rectangles

4

2x + 3

A = bh 8x 12

The total area of the large rectangle is 8x + 12.

FactoringHow can we work backwards from this process?

We need to start with our solution: 128 x

Think about the algebra tiles– can we build a rectangle?

Let’s look at the dimensions of the rectangle we made.

4

x x 1 1 1

2x + 3Simplify the base!

Area = (base)(height)

8x + 12 = (2x + 3)(4)** Don’t forget that using generic rectangles can save time!

FactoringExample 2: Using Generic Rectangles

xx 84 2 When asked to factor, you are given the area and are asked to find the dimensions.

24x x84x

When finding the dimensions you want to find the largest height possible. This height is the GREATEST COMMON FACTOR!

1) Find a height that works for both small rectangles.2) Use that height and the areas to find the base.

x + 2

The factored form: 4x(x+2)

PracticeUse the concept of area to find the factored form of each of the following expressions.

xx

xx

xx

93)4

105)3

1812)263)1

3

2

Solutions:

)3(3)4

)2(5)3)32(6)2

)2(3)1

2

xx

xxxx

Once you have completed all the problems,

click the mouse again for the

solutions!

Multiplication ReviewExample 2

)3)(5( xx•Again we are going to use the concept of area to multiply our 2 factors.

•Use each of the factors for either the base or the height.

** Because it is a multiplication problem, it does not matter which factor is the base and which is the height.

x

+

5

x + 3

•Multiply base and height to find the area.

2x x3

x5 15

158)3)(5( 2 xxxx

Don’t forget to combine like terms!

FactoringRemember that factoring is working backwards to write a multiplication problem.

When given an expression with 3 terms, there are 2 methods of factoring that can be used.

2092 xxMethod 1: Generic Rectangle

2x

20

x9______ 4x 5x

4x

5x

Once all of the areas are labeled, we have to find the dimensions.Label each of the individual rectangles, then find the dimensions of the larger rectangle.

xx

5

4

)4)(5(2092 xxxxWe use 4 & 5 because they also multiply to equal 20!

FactoringMethod 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.

This is the same process we used to complete diamond problems.

2092 xx

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.

2x

MA

20

9

4 5 )4)(5(2092 xxxx

PracticeUse either generic rectangles or diamonds to factor each of the following.

183)4

152)3

128)2

86)1

2

2

2

2

xx

xx

xx

xx Once you have factored all of these, click the mouse again to

check your answers!

)3)(6)(4)3)(5)(3)2)(6(2)4)(2)(1

xxxxxxxx

Remember, the order Remember, the order of the factors does not of the factors does not matter!matter!

Special CasesThere are 2 special cases for factoring: Difference of Squares and Perfect Square Trinomial.

Difference of SquaresExample:

162 xDifference of squares is used when you have 2 terms separated by subtraction.

1) Rewrite the expression with 3 terms.1602 xx

2) Use a diamond problem to factor.-16

0

-4 4 3) Write final factors.

)4)(4( xx

Special CasesPerfect Square Square Trinomials

36122 xxA perfect square trinomial looks just like a diamond problem. The difference is in how we write the answer.

1) Factor using a diamond.

36

126 6

)6)(6( xx2) What do you notice about your solution? Can we write it a simpler way?

2)6( x

PracticeIn each of the problems below, first decide whether you have a difference of squares or a perfect square trinomial, then factor.

25)4

20)3

209)2

9)1

2

2

2

2

x

xx

xx

x Solve each of the problems.

After you finish, click the mouse to check your answers.

)5)(5)(4)4)(5)(3)5)(4)(2)3)(3)(1

xxxxxxxx

Factoring CompletelyNow that we have talked about several methods of factoring, let’s put them together!!

Factoring completely is a combination of generic rectangles and diamond problems to present our answer in simplest form!

24 82 xx

1) Set up a generic rectangle to factor the GCF– find the largest height!

42x28x22x2) Find the base of the rectangle.

42 x)4(2 22 xx

3) Try to write the base with 3 terms to factor. )40(2 22 xxx

4) Now, we can use a diamond to factor the base.

-4

0

2 -25) Write your final expression.

)2)(2(2 2 xxx

An Exception– ax2

Not all expressions have a GCF.

273 2 xx

Note that the terms in the example do not have anything in common, but there is still a coefficient (3) in front of x2.

We need to use a modified diamond.

1. Multiply the 1st and 3rd terms. 623

2. Set up a diamond problem with the product in the top.

6

76 13. Rewrite the expression with

4 terms.2163 2 xxx

4. Use these terms to fill in the area of a generic rectangle.

23x x6

x1 25. Find the dimensions and write the factors.

x3

x 2

1)2)(13( xx

PracticeFactor each of the polynomials completely. Don’t forget to start with the GCF (largest height) first!

18122)4

6135)3

44)2

45243)1

2

2

2

2

xx

xx

x

xx After you factor each

problem, click the mouse

again to check your answers.

2)3(2)4

)3)(25)(3)1)(1(4)2)5)(3(3)1

x

xxxxxx

ReviewRemember: 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!

2 Terms: Difference of Squares

3 terms:

)5)(5(3)250(3

)25(3

753

2

2

2

xxxx

x

x

)3)(6(2)183(2

36622

2

xxxx

xx

Diamond Problem

Modified Diamond

)2)(35(63105

3065675

2

2

xxxxx

xx