1. Factoring

2. Factoring expressions
Factoring an expression is the opposite of multiplying.
Multiplying
Factoring
a(b + c)
ab + ac
Often:
When we multiply an expression we remove the parentheses.
When we factor an expression we write it with parentheses.

3. Factoring expressions
Expressions can be factored by dividing each term by a common factor and writing this outside of a pair of parentheses.
For example, in the expression
5x + 10
the terms 5x and 10 have a common factor, 5.
We can write the 5 outside of a set of parentheses and mentally divide 5x + 10 by 5.
Encourage students to check this by multiplying the expression to 5x + 10.
(5x + 10) ÷ 5 = x + 2
This is written inside the parentheses.
5(x + 2)
5(x + 2)

4. Factoring expressions
Writing 5x + 10 as 5(x + 2) is called factoring the expression.
Factor 6a + 8
Factor 12n – 9n2
The greatest common factor of 6a and 8 is
The greatest common factor of 12n and 9n2 is 2.
3n.
(6a + 8) ÷ 2 =
3a + 4
(12n – 9n2) ÷ 3n =
4 – 3n
Point out that we do not normally show the line involving division. This is done mentally. We can check the answer by using the distributive property.
6a + 8 =
2(3a + 4)
12n – 9n2 =
3n(4 – 3n)

5. Factoring expressions
Writing 5x + 10 as 5(x + 2) is called factoring the expression.
Factor 3x + x2
Factor 2p + 6p2 – 4p3
The greatest common factor of 3x and x2 is
The greatest common factor of 2p, 6p2 and 4p3 is x.
2p.
(2p + 6p2 – 4p3) ÷ 2p =
(3x + x2) ÷ x =
3 + x
1 + 3p – 2p2
3x + x2 =
x(3 + x)
2p + 6p2 – 4p3 =
2p(1 + 3p – 2p2)

6. Factoring
Start by asking students to give you the value of the greatest common factor of the two terms. Reveal this and then ask students to give you the values of the terms inside the parentheses.

7. Factoring by grouping
Some expressions containing four terms can be factored by regrouping the terms into pairs that share a common factor. For example:
Factor 4a + ab b
Two terms share a common factor of 4 and the remaining two terms share a common factor of b.
4a + ab b = 4a ab + b
= 4(a + 1) + b(a + 1)
4(a + 1) and + b(a + 1) share a common factor of (a + 1) so we can write this as
(a + 1)(4 + b)

8. When we take out a factor
Factoring by pairing
Factor xy – 6 + 2y – 3x
We can regroup the terms in this expression into two pairs of terms that share a common factor.
When we take out a factor
of –3, – 6 becomes + 2
xy – 6 + 2y – 3x = xy + 2y – 3x – 6
= y(x + 2) – 3(x + 2)
This expression could also be written as xy – 3x + 2y – 6 to give x(y – 3) + 2(y – 3) = (y – 3)(x + 2)
y(x + 2) and – 3(x + 2) share a common factor of (x + 2) so we can write this as
(x + 2)(y – 3)