Factoring – GCF, Grouping, & a = 1FRIDAY – January 31st,

2014

Warm Up

Find the greatest common factor: 72 and 96

Simplify

Simplify

3/1)8( 62 yx

2

2

4

18

y

y

Greatest Common Factor (GCF) The Greatest Common Factor (GCF) of two or more

numbers is the largest number that can divide into all of the numbers.

To find the GCF, start by writing out the prime factorization. What factors do the numbers have in common? Circle these then multiply the common factors to get your answer.

We can also find the GCF of monomials…

Example 1:

Find the GCF of 40a2b and 48ab4

Examples

2. Find the GCF of 12a3b4 and 3a5b.

3. Find the GCF of 7x2y2 and 10xy3.

Undoing?!?!?

What operation undoes addition for numbers?

What operation undoes division for numbers?

How can we undo multiplication for polynomials?

Remember this warm up problem?

Multiply: 2x(5x + 3)

We can work backwards. What if I gave you the answer 10x2 + 6x, and asked you for the original problem?

We can also “undistribute” our warm up problem

Find the GCF then use the distributive property to factor out the GCF.

What are we really doing? Take out the GCF and then write your “leftovers” on the inside.

Example 13: 10x2 + 6x

We can also find the GCF of a polynomial…

Factor out the GCF of all of the terms.

Example 9: 3x3y – 9x2y2

Example 10: 12x3 – 8x2 + 16x

Examples 11 & 12

18z3 + 9z2 – 6z

12a2b + 24a – 48b

Examples

Factor out the GCF from each of the following:

14. 3x3y – 9x2y2 15. 12a2b + 24a – 48b

Practice

Factor out the GCF of each of the following:

24c5 – 16c2d

19x2y + 9xy2

32x3 – 4x – 16

Practice

Find the GCF of 120x2y5 and 60x4y5

A. 60x4y5

B. 60x2y5

C. 2x4y5

D. 2x2y2

Practice

What is the GCF of the terms of

A. 2 B. 4 C. 2c D. 4c

Practice

Don’t forget

to check your

answer!!

Practice

Factor 15x2 -12x + 5

A. 5(3x2 – 2x + 1)B. x(15x2 – 12x + 5)C. (x + 3)(x – 5)D.Prime

Practice

Factor 12x3y2 – 9x2y4 + 6x2y.

A. 3x2y(4xy – 3y3 + 2)B. 3x2y(4x – 3y3 + 2y)C.xy(12x2y – 9xy3 + 6x)D.3xy(4x2y – 3xy3 + 2x)

Factoring by Grouping

You can try to factor by grouping when your polynomial has FOUR terms.

1. Always look for a GCF first!2. Then group your terms in pairs using

parenthesis.3. Find the GCF for each binomial.4. Your answer will be (GCF’s)(leftovers).

***Your leftovers must match***

Example #1

Always look for a GCF first!

Then group your terms in pairs using parenthesis.

Find the GCF for each binomial.

Your answer will be (GCF’s)(leftovers).

Your leftovers must match

Factor: 12x3 + 3x2 + 20x + 5

How can we check our answers?

Example #2

Factor: 8x2 + 8xy + 2y2 + 2xy

Don’t forget to look for a

GCF first!

Examples 3 & 4:

8j3 + 4j2 + 10j + 5 2m3 + 8m2 + 9m + 36

THINK/PAIR/SHARE

Where did the teacher make an error?

Factor 20p3 + 40p2 + 15p + 30

(20p3 + 40p2) + (15p + 30)

20p2(p + 2) + 15(p + 2)

(p + 2)(20p2+15)

Factoring trinomial with a = 1

Solve the x-puzzles!

Factoring with a = 1

Start with x2 + bx + c FIND the 2 factors!

Example 1: 1. x2 + 11x + 18 We need 2

#’s whose product is 18 and sum is 11

Examples

2. Factor. a2 – 9a + 20

3. Factor. c2 – c – 30

Patterns in Factoring

If c is positive (x2 + bx + c), then both binomials will have the sign of b.

If c is negative (x2 + bx – c), then you will have one binomial with addition, and one with subtraction.

Examples

4. Factor x2 - 4x – 21

5. Factor x2 - 8x + 7

How do I know what to do when?

Always check for a GCF! And if there is one, factor (un-distribute) it FIRST.

After it’s factored out or if there isn’t one, see how many terms it has.

4 terms factor by grouping

3 terms factor the trinomial (like we did today)

Homework

Worksheet