Is x - 5 a factor of x^3 - 4x^2 - 7x + 10 Step by Step

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The question

Question: is (x − 5) a factor of x³ − 4x² − 7x + 10?

(x − 5) and x³ − 4x² − 7x + 10

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 − 4x² − 7x + 10

2 Make x − 5 zero

Which x makes x − 5 zero?

x − 5 = 0

5 take away 5 is 0.

So x = 5.

x = 5

Careful!

Many people put in −5, the number they see.

But x − 5 is 0 when x is 5.

Careful: not f(−5) ✗

3 Work out f(5)

Put 5 in place of every x.

We write it in brackets.

f(5) = 5³ − 4(5)² − 7(5) + 10

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

= 125 − 100 − 35 + 10

Then add and take away from left to right.

f(5) = 0.

f(5) = 0

4 Read the answer

So the remainder is 0.

remainder = 0

Nothing is left over.

So (x − 5) is a factor!

5 Check

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

We use 5 again.

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

The same as f(5)!

Last number: 0 = f(5)

The other numbers give the other factor.

So x³ − 4x² − 7x + 10 is (x − 5)(x² + x − 2).

x³ − 4x² − 7x + 10 = (x − 5)(x² + x − 2)

Let's look back.

First: call it f(x).

Then: make x − 5 zero.

Then: work out f(5).

Then: read the answer.

Last: check.

Answer