Worked examples
The question
Question: simplify (x² − 9)/(x + 3).
A fraction with x in it is called a rational expression.
x2 − 9x + 3
We make it simpler like a number fraction.
6/9: both have a 3 in them.
3/3 is 1, so 2/3 is left.
We do the same with brackets.
69 = 2 × 33 × 3 = 23
1 Factor the top
We look at the top: x² − 9.
= x2 − 9
x² − 9 is x² minus 3 squared.
A square minus a square is the plus bracket times the minus bracket.
a² − b² = (a + b)(a − b)
So the top is (x + 3)(x − 3).
= (x + 3)(x − 3)
2 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.
Careful: only whole brackets cross out
Now we write the fraction with the factors: (x + 3)(x − 3)/(x + 3).
= (x + 3)(x − 3)(x + 3)
(x + 3)/(x + 3) is 1.
Cross them out.
3 Write what is left
What is left is x − 3.
= x − 3
One more thing: the bottom of the question can't be 0.
So x can't be −3.
x ≠ −3
4 Check
Let's check with a number: x = 2.
First in the question.
We get −1.
Question: 22 − 92 + 3 = −55 = −1
Now in our answer.
We get −1 too.
Answer: 2 − 3 = −1
Both are −1.
So x − 3 is right!
Let's look back.
First: factor the top.
Then: cross out the same.
Then: write what is left.
Last: check.
▶ Watch it solved
The question
Question: simplify (2x + 6)/(x² + 5x + 6).
A fraction with x in it is called a rational expression.
2x + 6x2 + 5x + 6
We make it simpler like a number fraction.
6/9: both have a 3 in them.
3/3 is 1, so 2/3 is left.
We do the same with brackets.
69 = 2 × 33 × 3 = 23
1 Factor the top
We look at the top: 2x + 6.
= 2x + 6
2 and 6 can both be divided by 2.
We take 2 out.
What is left goes in the bracket.
2x ÷ 2 = x, 6 ÷ 2 = 3
So the top is 2(x + 3).
= 2(x + 3)
2 Factor the bottom
We look at the bottom: x² + 5x + 6.
= x2 + 5x + 6
For x² + 5x + 6, we need two numbers.
They multiply to 6, and they add to 5.
? × ? = 6, ? + ? = 5
They multiply to a plus number, so the two numbers have the same sign.
They add to a plus number, so both are plus.
× is + → same signs: both +
We list the pairs that make 6, smallest first.
We add each pair.
We need 5.
2 and 3 give 5.
Yes!
1 × 6: 1 + 6 = 7
2 × 3: 2 + 3 = 5 ✓
So the bottom is (x + 2)(x + 3).
= (x + 2)(x + 3)
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.
Careful: only whole brackets cross out
Now we write the fraction with the factors: 2(x + 3)/(x + 2)(x + 3).
= 2(x + 3)(x + 2)(x + 3)
(x + 3)/(x + 3) is 1.
Cross them out.
4 Write what is left
What is left is 2/(x + 2).
= 2x + 2
One more thing: the bottom of the question can't be 0.
So x can't be −3 or −2.
x ≠ −3, −2
5 Check
Let's check with a number: x = 2.
First in the question.
We get 1/2.
Question: 2(2) + 622 + 5(2) + 6 = 1020 = 12
Now in our answer.
We get 1/2 too.
Answer: 22 + 2 = 12
Both are 1/2.
So 2/(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.
▶ Watch it solved
The question
Question: simplify (3 − x)/(x − 3).
A fraction with x in it is called a rational expression.
3 − xx − 3
We make it simpler like a number fraction.
6/9: both have a 3 in them.
3/3 is 1, so 2/3 is left.
We do the same with brackets.
69 = 2 × 33 × 3 = 23
1 Factor the top
We look at the top: 3 − x.
= 3 − x
The x part has a minus.
We take the minus out, in front of a bracket.
Every sign inside changes.
3 − x = −(x − 3)
So the top is −(x − 3).
= −(x − 3)
2 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.
Careful: only whole brackets cross out
Now we write the fraction with the factors: −(x − 3)/(x − 3).
= −(x − 3)(x − 3)
(x − 3)/(x − 3) is 1.
Cross them out.
3 Write what is left
What is left is −1.
= −1
One more thing: the bottom of the question can't be 0.
So x can't be 3.
x ≠ 3
4 Check
Let's check with a number: x = 2.
First in the question.
We get −1.
Question: 3 − 22 − 3 = 1−1 = −1
Now in our answer.
We get −1 too.
Answer: −1
Both are −1.
So −1 is right!
Let's look back.
First: factor the top.
Then: cross out the same.
Then: write what is left.
Last: check.
▶ Watch it solved
