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

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The question

Question: is (x + 2) a factor of x³ + x² − 5?

(x + 2) and x³ + x² − 5

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 + x² − 5

2 Make x + 2 zero

Which x makes x + 2 zero?

x + 2 = 0

x + 2 is the same as x minus −2.

So c is −2.

x + 2 = x − (−2)

−2 plus 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.

We write it in brackets.

f(−2) = (−2)³ + (−2)² − 5

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

= −8 + 4 − 5

Then add and take away from left to right.

f(−2) = −9.

f(−2) = −9

4 Read the answer

So the remainder is −9.

It is not 0.

remainder = −9

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: −9.

The same as f(−2)!

Last number: −9 = 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