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

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The question

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

2x³ − 3x² + x − 4x − 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.

divide by x − c → remainder = f(c)

1 Call it f(x)

Let's call the polynomial f(x).

f(x) = 2x³ − 3x² + x − 4

2 Make x − 2 zero

Which x makes x − 2 zero?

x − 2 = 0

2 take away 2 is 0.

So x = 2.

x = 2

Careful!

Many people put in −2, the number they see.

But x − 2 is 0 when x is 2.

Careful: not f(−2) ✗

3 Work out f(2)

Put 2 in place of every x.

We write it in brackets.

f(2) = 2(2)³ − 3(2)² + 2 − 4

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

= 16 − 12 + 2 − 4

Then add and take away from left to right.

f(2) = 2.

f(2) = 2

4 Read the remainder

So the remainder is 2.

remainder = 2

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: 2.

The same as f(2)!

Last number: 2 = 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.

Answer