ObjectivesThe student will be able to:

Factor using the greatest common factor (GCF).

Lesson 4-4

Review: What is the GCF of 25a2 and 15a?

5a

Let’s go one step further…

1) FACTOR 25a2 + 15a.

Find the GCF and divide each term

25a2 + 15a = 5a( ___ + ___ )

Check your answer by distributing.

225

5

a

a

15

5

a

a

5a 3

2) Factor 18x2 - 12x3.Find the GCF

6x2

Divide each term by the GCF

18x2 - 12x3 = 6x2( ___ - ___ )

Check your answer by distributing.

2

2

18

6

x

x

3

2

12

6

x

x

3 2x

3) Factor 28a2b + 56abc2.GCF = 28ab

Divide each term by the GCF28a2b + 56abc2 = 28ab ( ___ + ___ )

Check your answer by distributing.28ab(a + 2c2)

228

28

a b

ab

256

28

abc

ab

a 2c2

Factor 20x2 - 24xy

1. x(20 – 24y)

2. 2x(10x – 12y)

3. 4(5x2 – 6xy)

4. 4x(5x – 6y)

5) Factor 28a2 + 21b - 35b2c2

GCF = 7Divide each term by the GCF

28a2 + 21b - 35b2c2 = 7 ( ___ + ___ - ____ )

Check your answer by distributing.7(4a2 + 3b – 5b2c2)

228

7

a 21

7

b

4a2 5b2c2

2 235

7

b c

3b

Factor 16xy2 - 24y2z + 40y2

1. 2y2(8x – 12z + 20)

2. 4y2(4x – 6z + 10)

3. 8y2(2x - 3z + 5)

4. 8xy2z(2 – 3 + 5)

Factoring Polynomials

Grouping, Trinomials, Binomials, GCF & Solving Equations

Factor by Grouping

When polynomials contain four terms, it is sometimes easier to group like terms in order to factor.

Your goal is to create a common factor. You can also move terms around in the

polynomial to create a common factor. Practice makes you better in recognizing

common factors.

Factoring Four Term Polynomials

Factor by GroupingExample 1:

FACTOR: 3xy - 21y + 5x – 35 Factor the first two terms: 3xy - 21y = 3y (x – 7) Factor the last two terms: + 5x - 35 = 5 (x – 7) The green parentheses are the same so it’s the

common factor Now you have a common factor

(x - 7) (3y + 5)

Factor by Grouping Example 2: FACTOR: 6mx – 4m + 3rx – 2r Factor the first two terms: 6mx – 4m = 2m (3x - 2) Factor the last two terms: + 3rx – 2r = r (3x - 2) The green parentheses are the same so it’s the

common factor Now you have a common factor

(3x - 2) (2m + r)

Factor by Grouping Example 3:

FACTOR: 15x – 3xy + 4y –20 Factor the first two terms: 15x – 3xy = 3x (5 – y) Factor the last two terms: + 4y –20 = 4 (y – 5) The green parentheses are opposites so change

the sign on the 4 - 4 (-y + 5) or – 4 (5 - y) Now you have a common factor (5 – y) (3x – 4)

Factoring Trinomials

Factoring Trinominals

1. When trinomials have a degree of “2”, they are known as quadratics.

2. We learned earlier to use the “diamond” to factor trinomials that had a “1” in front of the squared term.

x2 + 12x + 35

(x + 7)(x + 5)

More Factoring Trinomials

3. When there is a coefficient larger than “1” in front of the squared term, we can use a modified diamond or square to find the factors.

3. Always remember to look for a GCF before you do ANY other factoring.

More Factoring Trinomials

5. Let’s try this example3x2 + 13x + 4

Make a box

Write the factors of the first term.Write the factors of the last term.Multiply on the diagonal and add to see if

you get the middle term of the trinomial. If so, you’re done!

Difference of Squares

Difference of Squares When factoring using a difference of

squares, look for the following three things:– only 2 terms– minus sign between them– both terms must be perfect squares

If all 3 of the above are true, write two ( ), one with a + sign and one with a – sign :

( + ) ( - ).

Try These

1. a2 – 16 2. x2 – 25 3. 4y2 – 16 4. 9y2 – 25 5. 3r2 – 81 6. 2a2 + 16

Perfect Square Trinomials

Perfect Square Trinomials When factoring using perfect square

trinomials, look for the following three things:– 3 terms– last term must be positive– first and last terms must be perfect

squares If all three of the above are true, write one (

)2 using the sign of the middle term.

Try These

1. a2 – 8a + 16 2. x2 + 10x + 25 3. 4y2 + 16y + 16 4. 9y2 + 30y + 25 5. 3r2 – 18r + 27 6. 2a2 + 8a - 8

Factoring Completely

Factoring Completely Now that we’ve learned all the types of

factoring, we need to remember to use them all.

Whenever it says to factor, you must break down the expression into the smallest possible

factors.

Let’s review all the ways to factor.

Types of Factoring1. Look for GCF first.2. Count the number of terms:

a) 4 terms – factor by groupingb) 3 terms -

1. look for perfect square trinomial2. if not, try diamond or box

c) 2 terms - look for difference of squares

If any ( ) still has an exponent of 2 or more, see if you can factor again.

Solving Equations by Factoring

Steps to Solve Equations by Factoring

1. We know that an equation must be solved for the unknown.

2. Up to now, we have only solved equations with a degree of 1.

2x + 8 = 4x +6

-2x + 8 = 6

-2x = -2

x = 1

Steps to Solve Equations by Factoring

3. If an equation has a degree of 2 or higher, we cannot solve it until it has been factored.

4. You must first get “0” on one side of the = sign before you try any factoring.

5. Once you have “0” on one side, use all your rules for factoring to make 2 ( ) or factors.

Steps to Solve Equations by Factoring

6. Next, set each factor = 0 and solve for the unknown.

x2 + 12x = 0 Factor GCF

x(x + 12) = 0

(set each factor = 0, & solve)

x = 0 x + 12 = 0

x = - 12

7. You now have 2 answers, x = 0 and x = -12.