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

Step by step

The question

Question: find the remainder when x³ − 2x² + 4 is divided by (x − 3).

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 − 3 zero

Which x makes x − 3 zero?

3 take away 3 is 0.

So x = 3.

Careful!

Many people put in −3, the number they see.

But x − 3 is 0 when x is 3.

3 Work out f(3)

Put 3 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(3) = 13.

4 Read the remainder

So the remainder is 13.

5 Check

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

We use 3 again.

The last number in the table is the remainder: 13.

The same as f(3)!

Let's look back.

First: call it f(x).

Then: make x − 3 zero.

Then: work out f(3).

Then: read the remainder.

Last: check.