Is x - 3 a factor of x^3 - 27 Step by Step

Step by step

The question

Question: is (x − 3) a factor of x³ − 27?

(x − 3) and x³ − 27

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 − 27

2 Make x − 3 zero

Which x makes x − 3 zero?

x − 3 = 0

3 take away 3 is 0.

So x = 3.

x = 3

Careful!

Many people put in −3, the number they see.

But x − 3 is 0 when x is 3.

Careful: not f(−3) ✗

3 Work out f(3)

Put 3 in place of every x.

f(3) = 3³ − 27

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

= 27 − 27

Then add and take away from left to right.

f(3) = 0.

f(3) = 0

4 Read the answer

So the remainder is 0.

remainder = 0

Nothing is left over.

So (x − 3) is a factor!

5 Check

Let's check with synthetic division, from the division lesson.

We use 3 again.

The last number in the table is the remainder: 0.

The same as f(3)!

Last number: 0 = f(3)

The other numbers give the other factor.

So x³ − 27 is (x − 3)(x² + 3x + 9).

x³ − 27 = (x − 3)(x² + 3x + 9)

Let's look back.

First: call it f(x).

Then: make x − 3 zero.

Then: work out f(3).

Then: read the answer.

Last: check.

Answer