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

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The question

Question: is (x − 1) a factor of x² + 2x − 3?

(x − 1) and x² + 2x − 3

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 + 2x − 3

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.

We write it in brackets.

f(1) = 1² + 2(1) − 3

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

= 1 + 2 − 3

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 x² + 2x − 3 is (x − 1)(x + 3).

x² + 2x − 3 = (x − 1)(x + 3)

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