Remainder when 2x^4 - x^3 + 3x - 5 is divided by x + 1 Step by Step

Step by step

The question

Question: find the remainder when 2x⁴ − x³ + 3x − 5 is divided by (x + 1).

2x⁴ − x³ + 3x − 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) = 2x⁴ − x³ + 3x − 5

2 Make x + 1 zero

Which x makes x + 1 zero?

x + 1 = 0

x + 1 is the same as x minus −1.

So c is −1.

x + 1 = x − (−1)

−1 plus 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.

We write it in brackets.

f(−1) = 2(−1)⁴ − (−1)³ + 3(−1) − 5

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

= 2 + 1 − 3 − 5

Then add and take away from left to right.

f(−1) = −5.

f(−1) = −5

4 Read the remainder

So the remainder is −5.

remainder = −5

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: −5.

The same as f(−1)!

Last number: −5 = 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