Remainder when x^3 + x - 5 is divided by x

Step by step

The question

Question: find the remainder when x³ + x − 5 is divided by (x − 1).

x³ + x − 5x − 1

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 + x − 5

2 Make x − 1 zero

Which x makes x − 1 zero?

x − 1 = 0

1 take away 1 is 0.

So x = 1.

x = 1

Careful!

Many people put in −1, the number they see.

But x − 1 is 0 when x is 1.

Careful: not f(−1) ✗

3 Work out f(1)

Put 1 in place of every x.

f(1) = 1³ + 1 − 5

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

= 1 + 1 − 5

Then add and take away from left to right.

f(1) = −3.

f(1) = −3

4 Read the remainder

So the remainder is −3.

remainder = −3

5 Check

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

We use 1 again.

The last number in the table is the remainder: −3.

The same as f(1)!

Last number: −3 = f(1)

Let's look back.

First: call it f(x).

Then: make x − 1 zero.

Then: work out f(1).

Then: read the remainder.

Last: check.

Answer