Factoring Expressions

Definition

Factoring is the process of finding an alternative or shortened way to write an expression, with it still remaining equivalent.

Factor these expressions by writing them as products.

The solutions in the above animation are repeated with a more detailed explanation below.

Question 1.1

3x + 24

In factoring, we will often look for what is called a “common factor”. This means a number multiplicatively “inside” two or more numbers. You can take that common factor and multiply it by two other numbers, and it will give you the original numbers. For example, 4 & 6 both have a common factor of 2. 4 can be written as 2 x 2 and 6 can be written as 2 x 3, we can see two is common in both those products, and is what is called a common factor.

So, we want to find the common factor in our expression 3x + 24,

A good method for this, is to take the smallest number in the expression, and look at what numbers could be multiplied to get that term AS WELL AS the other term.

So, for the smallest term, we have 3x, we know that 3 could be multiplied by x to give us 3x, lets see if 3 multiplied by something could give us 24.

So, divide 24 ÷ 3 = 8, so 3 x 8 = 24, now let's write our expression now with the original terms expressed as products of 3. So:

3x + 24 =

3(x) + 3(8), if we recognize the distributive property here then we are on the right track!

Now try and imagine going back one step from the distributive property,

3(x + 8).

And there we have our factored expression.

Question 1.2

-2x + 14

Same as above, think about what one term could be multiplied by two different terms to give us the above expression,

OR

What term divides into both terms in the expression?

That would be 2

-2x = 2(-x) & 14 = 2(7), so,

-2x + 14 =

2(-x) + 2(7) =

Opposite Distributive property

2(-x + 7) or 2(7 - x)

Question 1.3

7x - 7

Look for what divides into both terms, as there is 7 in both terms we simply choose 7,

7x - 7 =

7(x) - 7(1) =

Don't remove the (1) term when removing the common factor or 7! Its division, not subtraction

Opposite Distributive property,

7(x - 1)

Question 1.4

Expand and simplify by removing a common factor for this expression:

x + 3y + x + 3 + x

So, first, we see our three x terms, so we group them together, using the commutative property.

x + x + x + 3y + 3

Now rewrite x + x + x to 3x,

3x + 3y + 3,

Note we now have a common factor of 3 amongst our three terms, so we can remove that 3 using “opposite distributive property”

3(x + y + 1)