Is x + 1 a factor of x^3 + 2x^2 - x

Step by step

The question

Question: is (x + 1) a factor of x³ + 2x² − x − 2?

(x + 1) and x³ + 2x² − x − 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.

The factor theorem: if the remainder is 0, it goes in exactly.

Then it is a factor.

divide by x − c → remainder = f(c)

remainder 0 → a factor

1 Call it f(x)

Let's call the polynomial f(x).

f(x) = x³ + 2x² − x − 2

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) = (−1)³ + 2(−1)² − (−1) − 2

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

= −1 + 2 + 1 − 2

Then add and take away from left to right.

f(−1) = 0.

f(−1) = 0

4 Read the answer

So the remainder is 0.

remainder = 0

Nothing is left over.

So (x + 1) is a factor!

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: 0.

The same as f(−1)!

Last number: 0 = f(−1)

The other numbers give the other factor.

So x³ + 2x² − x − 2 is (x + 1)(x² + x − 2).

x³ + 2x² − x − 2 = (x + 1)(x² + x − 2)

Let's look back.

First: call it f(x).

Then: make x + 1 zero.

Then: work out f(−1).

Then: read the answer.

Last: check.

Answer