Factoring by Grouping

When you have 4 terms

• Remember– To factor out a GCF, we write the GCF on the outside and divide each term by the GCF.

• Ex) x5y – x3y + 3y = y(x5 – x3 + 3)

• You try: 3x2y - 9xy + 18xy2

3xy(x – 3 + 6y)

• So, if instead of a variable, our GCF is a binomial, we do the same thing:

3(x – 2) + 5y(x – 2)

Our GCF is: (x – 2)

When we pull that out we get…

(x – 2) (3 + 5y)

You try:

• x(a – 4) – 3( a – 4)

Did you get: (a - 4)(x - 3)???

m(p + 3) + q(p + 3)

Did you get: (p + 3)(m + q)???

What if the GCF is not obvious?• What if I have a problems like:

x2 + 3x + 2xy + 6y

That is when we have to group terms together. Always try to group the 1st 2 and the last 2.

x2 + 3x + 2xy + 6y

x(x + 3) + 2y(x + 3)

(x + 3)(x + 2y)

Take out the GCF of each group

What is in parentheses will be the same (most of the time)

Let’s try:

1) y2 – 5wy + 4y - 20w

y (y – 5w) + 4(y – 5w)

(y – 5w)(y + 4)

2) ab + 7b – 3a – 21

b(a + 7) – 3(a + 7)

(a + 7)(b – 3)

If the middle sign is a negative, you must remove a negative from the second set. Remember, when you divide a number by a negative, the sign changes

What if what is in my parentheses is not the same???

x2 – 5x3 + 15x – 3x2(1 – 5x) + 3(5x – 1)x2(1 – 5x) - 3(1 – 5x)(1 – 5x)(x2 – 3)

You try: ax - bx + by – ay x(a - b) + y (b - a)

x(a - b) – y (a – b) (a – b) (x – y)

Is -1(5x – 1) the same thing as (1 -5x)????

Yes!!!! So, change that +3 to a -3 and then the insides of the ( ) will be the same thing!!

Try these:

1. 5xy2 – 20x + 3y3 – 12y

2. 2x3 + 10x – x2y - 5y

3. 12n3 + 15n2 + 4n + 5

Answers: 1. (5x + 3y)(y2 – 4)

(5x + 3y)(y + 2)(y – 2)

2. (2x – y)(x2 + 5) 3. (3n2 + 1)(4n + 5)

Homework

• 1. 2x3 + 4x2 + x + 2

• 2. 2x3 + 6x2 + 3x + 9

• 3. 2x3 + 12x2 – 5x – 30

• 4. 9x3 – 12x2 – 24x + 18