Worked examples
The question
Question: divide (x² + 5x + 6) by (x + 2).
x2 + 5x + 6x + 2
1 Set it up
We write it like long division with numbers.
x + 2 goes outside, x² + 5x + 6 inside.
2 First part of the answer
Divide the first parts: x² divided by x is x.
We write it on top.
Multiply: x times (x + 2) is x² + 2x.
We write it under, with a minus: we take it away.
x² take away x² is 0.
That is why we chose x.
5x take away 2x is 3x.
Bring down the next part: 6.
3 Second part of the answer
Divide the first parts: 3x divided by x is 3.
We write it on top.
Multiply: 3 times (x + 2) is 3x + 6.
We write it under, with a minus: we take it away.
3x take away 3x is 0.
That is why we chose 3.
6 take away 6 is 0.
The last part is 0.
So nothing is left over.
So the answer is x + 3.
Nothing is left over.
x2 + 5x + 6x + 2 = x + 3
4 Check
Let's check with a number: x = 2.
First the start: x² + 5x + 6.
Start: 22 + 5(2) + 6 = 4 + 10 + 6 = 20
Now the answer: (x + 2) times (x + 3).
Answer: (2 + 2) × (2 + 3) = (4) × (5) = 20
Both give 20.
So the answer is right!
Both give 20
Let's look back.
First: set it up.
Then: first part of the answer.
Then: second part of the answer.
Last: check.
Answer x2 + 5x + 6x + 2 = x + 3
▶ Watch it solved
The question
Question: divide (x³ − 2x² − 5x + 6) by (x − 3).
x3 − 2x2 − 5x + 6x − 3
1 Set it up
We write it like long division with numbers.
x − 3 goes outside, x³ − 2x² − 5x + 6 inside.
2 First part of the answer
Divide the first parts: x³ divided by x is x².
We write it on top.
Multiply: x² times (x − 3) is x³ − 3x².
We write it under, with a minus: we take it away.
x³ take away x³ is 0.
That is why we chose x².
−2x² take away −3x² is x².
Bring down the next part: −5x.
3 Second part of the answer
Divide the first parts: x² divided by x is x.
We write it on top.
Multiply: x times (x − 3) is x² − 3x.
We write it under, with a minus: we take it away.
x² take away x² is 0.
That is why we chose x.
−5x take away −3x is −2x.
Bring down the next part: 6.
4 Third part of the answer
Divide the first parts: −2x divided by x is −2.
We write it on top.
Multiply: −2 times (x − 3) is −2x + 6.
We write it under, with a minus: we take it away.
−2x take away −2x is 0.
That is why we chose −2.
6 take away 6 is 0.
The last part is 0.
So nothing is left over.
So the answer is x² + x − 2.
Nothing is left over.
x3 − 2x2 − 5x + 6x − 3 = x2 + x − 2
5 Check
Let's check with a number: x = 2.
First the start: x³ − 2x² − 5x + 6.
Start: 23 − 2(2)2 − 5(2) + 6
= 8 − 8 − 10 + 6 = −4
Now the answer: (x − 3) times (x² + x − 2).
Answer: (2 − 3) × (22 + 2 − 2) = (−1) × (4)
= −4
Both give −4.
So the answer is right!
Both give −4
Let's look back.
First: set it up.
Then: first part of the answer.
Then: second part of the answer.
Then: third part of the answer.
Last: check.
Answer x3 − 2x2 − 5x + 6x − 3 = x2 + x − 2
▶ Watch it solved
The question
Question: divide (x³ + 2x² − x + 5) by (x + 3).
x3 + 2x2 − x + 5x + 3
1 Set it up
We write it like long division with numbers.
x + 3 goes outside, x³ + 2x² − x + 5 inside.
2 First part of the answer
Divide the first parts: x³ divided by x is x².
We write it on top.
Multiply: x² times (x + 3) is x³ + 3x².
We write it under, with a minus: we take it away.
x³ take away x³ is 0.
That is why we chose x².
2x² take away 3x² is −x².
Bring down the next part: −x.
3 Second part of the answer
Divide the first parts: −x² divided by x is −x.
We write it on top.
Multiply: −x times (x + 3) is −x² − 3x.
We write it under, with a minus: we take it away.
−x² take away −x² is 0.
That is why we chose −x.
−x take away −3x is 2x.
Bring down the next part: 5.
4 Third part of the answer
Divide the first parts: 2x divided by x is 2.
We write it on top.
Multiply: 2 times (x + 3) is 2x + 6.
We write it under, with a minus: we take it away.
2x take away 2x is 0.
That is why we chose 2.
5 take away 6 is −1.
−1 has no x.
We can't divide it by x.
So −1 is the remainder: what is left over.
So the answer is x² − x + 2, with −1 left over: minus 1/(x + 3).
x3 + 2x2 − x + 5x + 3 = x2 − x + 2 − 1x + 3
5 Check
Let's check with a number: x = 2.
First the start: x³ + 2x² − x + 5.
Start: 23 + 2(2)2 − 2 + 5 = 8 + 8 − 2 + 5
= 19
Now the answer: (x + 3) times (x² − x + 2), minus 1.
Answer: (2 + 3) × (22 − 2 + 2) − 1
= (5) × (4) − 1 = 19
Both give 19.
So the answer is right!
Both give 19
Let's look back.
First: set it up.
Then: first part of the answer.
Then: second part of the answer.
Then: third part of the answer.
Last: check.
Answer x3 + 2x2 − x + 5x + 3 = x2 − x + 2 − 1x + 3
▶ Watch it solved
