Remainder when x^3 - 4x + 1 is divided by x + 2

Step by step

The question

Question: find the remainder when x³ − 4x + 1 is divided by (x + 2).

The remainder theorem: when we divide by x minus a number, the remainder is what f gives for that number.

We don't need to divide.

1 Call it f(x)

Let's call the polynomial f(x).

2 Make x + 2 zero

Which x makes x + 2 zero?

x + 2 is the same as x minus −2.

So c is −2.

−2 plus 2 is 0.

So x = −2.

Careful!

Many people put in 2, the number they see.

But x + 2 is 0 when x is −2.

3 Work out f(−2)

Put −2 in place of every x.

We write it in brackets.

Work out each part: the powers first, then the times.

Then add and take away from left to right.

f(−2) = 1.

4 Read the remainder

So the remainder is 1.

5 Check

Let's check with synthetic division, from the division lesson.

We use −2 again.

The last number in the table is the remainder: 1.

The same as f(−2)!

Let's look back.

First: call it f(x).

Then: make x + 2 zero.

Then: work out f(−2).

Then: read the remainder.

Last: check.