Is x - 2 a factor of x^2 + x + 1 Step by Step

Step by step

The question

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

(x − 2) and x² + x + 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.

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) = x² + x + 1

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.

f(2) = 2² + 2 + 1

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

= 4 + 2 + 1

Then add and take away from left to right.

f(2) = 7.

f(2) = 7

4 Read the answer

So the remainder is 7.

It is not 0.

remainder = 7

So (x − 2) is not a factor.

It does not go in exactly.

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: 7.

The same as f(2)!

Last number: 7 = f(2)

Let's look back.

First: call it f(x).

Then: make x − 2 zero.

Then: work out f(2).

Then: read the answer.

Last: check.

Answer