Remainder when x^3 - 6x^2 + 11x - 6 is divided by x

Step by step

The question

Question: find the remainder when x³ − 6x² + 11x − 6 is divided by (x − 4).

x³ − 6x² + 11x − 6x − 4

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) = x³ − 6x² + 11x − 6

2 Make x − 4 zero

Which x makes x − 4 zero?

x − 4 = 0

4 take away 4 is 0.

So x = 4.

x = 4

Careful!

Many people put in −4, the number they see.

But x − 4 is 0 when x is 4.

Careful: not f(−4) ✗

3 Work out f(4)

Put 4 in place of every x.

We write it in brackets.

f(4) = 4³ − 6(4)² + 11(4) − 6

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

= 64 − 96 + 44 − 6

Then add and take away from left to right.

f(4) = 6.

f(4) = 6

4 Read the remainder

So the remainder is 6.

remainder = 6

5 Check

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

We use 4 again.

The last number in the table is the remainder: 6.

The same as f(4)!

Last number: 6 = f(4)

Let's look back.

First: call it f(x).

Then: make x − 4 zero.

Then: work out f(4).

Then: read the remainder.

Last: check.

Answer