Step by step
The question
Question: divide (x³ + 2x² − x + 5) by (x + 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).
5 Check
Let's check with a number: x = 2.
First the start: x³ + 2x² − x + 5.
Now the answer: (x + 3) times (x² − x + 2), minus 1.
Both give 19.
So the answer is right!
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.