Step by step
The question
Question: simplify (x² − 16)/(x² − 2x − 8).
A fraction with x in it is called a rational expression.
1 Factor the top
We look at the top: x² − 16.
Factor means: write it as a times, with brackets.
x² − 16: 16 is 4 times 4, so it is x² minus 4 squared.
A square minus a square is the plus bracket times the minus bracket.
So the top is (x + 4)(x − 4).
2 Factor the bottom
We look at the bottom: x² − 2x − 8.
We want x² − 2x − 8 as two brackets: (x + ?)(x + ?).
We need two numbers.
They multiply to −8, and they add to −2.
Look at the sign in front of 8: it is minus.
So one number is plus and one is minus.
Now the sign in front of 2x: it is minus.
So the bigger one is minus.
We list the pairs that make 8, smallest first.
For each pair, we check two things: they multiply to 8, and the bigger take away the smaller is 2.
1 × 8 = 8.
8 − 1 = 7.
Not 2.
2 × 4 = 8.
4 − 2 = 2.
Both work!
So the two numbers are 2 and 4.
With the signs: 2 and −4.
So the bottom is (x + 2)(x − 4).
3 Cross out the same
Careful!
We cross out only a whole bracket, or a whole x, that multiplies everything.
Never one part of a bracket.
A number check: 2 plus 3, all over 3, is 5/3.
Crossing out the 3s would give 2.
Wrong!
Now we write the fraction with the factors: (x + 4)(x − 4)/(x + 2)(x − 4).
(x − 4)/(x − 4) is 1.
Cross them out.
4 Write what is left
What is left is (x + 4)/(x + 2).
One more thing: the bottom of the question can't be 0: we can't divide by 0.
x + 2 is 0 when x is −2.
x − 4 is 0 when x is 4.
The answer keeps this rule, even after crossing out.
So x can't be −2 or 4.
5 Check
Let's check with a number: x = 2.
We pick a small number that makes no bottom 0.
First in the question.
We get 3/2.
Now in our answer.
We get 3/2 too.
Both are 3/2.
So (x + 4)/(x + 2) is right!
Let's look back.
First: factor the top.
Then: factor the bottom.
Then: cross out the same.
Then: write what is left.
Last: check.