Is x + 3 a factor of 2x^3 + 5x^2 - 4x

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The question

Question: is (x + 3) a factor of 2x³ + 5x² − 4x − 3?

(x + 3) and 2x³ + 5x² − 4x − 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) = 2x³ + 5x² − 4x − 3

2 Make x + 3 zero

Which x makes x + 3 zero?

x + 3 = 0

x + 3 is the same as x minus −3.

So c is −3.

x + 3 = x − (−3)

−3 plus 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.

We write it in brackets.

f(−3) = 2(−3)³ + 5(−3)² − 4(−3) − 3

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

= −54 + 45 + 12 − 3

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 2x³ + 5x² − 4x − 3 is (x + 3)(2x² − x − 1).

2x³ + 5x² − 4x − 3 = (x + 3)(2x² − x − 1)

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