What is the remainder when x^3 - 4x^2 + 2x + 5 is divided by x

Step by step

The question

Question: find the remainder when x³ − 4x² + 2x + 5 is divided by (x − 3).

x³ − 4x² + 2x + 5x − 3

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Call it f(x)

Let's call the polynomial f(x).

We want the remainder, without the long division.

f(x) = x³ − 4x² + 2x + 5

2 Make x − 3 zero

Which x makes x − 3 zero?

We will see why that x helps.

x − 3 = 0

Is x alone now?

Not yet: − 3 is still with x.

How do we remove it?

We add 3 on both sides.

x − 3 + 3 = 0 + 3

So x = 3.

Check: 3 take away 3 is 0.

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) ✗

So after step 2: we added 3, and now we have x = 3.

← step 2: added 3

3 Why f(3) is the remainder

Think of 7 divided by 2.

It is 3, with 1 left over.

So 7 is 2 times 3, plus 1.

7 = 2 × 3 + 1

Dividing f(x) works the same way.

f(x) is (x − 3) times the answer, plus the remainder.

We stop dividing when the part left has no x.

So the remainder is just a number.

f(x) = (x − 3) × answer + remainder

Now put in x = 3.

f(3) = (3 − 3) × answer + remainder

3 take away 3 is 0.

And 0 times the answer is 0.

= 0 × answer + remainder

So f(3) is just the remainder.

We don't need to divide!

f(3) = remainder

This is called the remainder theorem.

divide by x − 3 → remainder = f(3)

4 Work out f(3)

Put 3 in place of every x.

We write it in brackets.

f(3) = 3³ − 4(3)² + 2(3) + 5

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

= 27 − 36 + 6 + 5

27 − 36 = −9, −9 + 6 = −3, −3 + 5 = 2

Then add and take away from left to right.

f(3) = 2.

f(3) = 2

5 Read the remainder

So the remainder is 2.

remainder = 2

6 Check

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

We use 3 again.

The last number in the table is the remainder: 2.

The same as f(3)!

Last number: 2 = f(3)

7 Pick the choice

Our answer is remainder = 2.

That is choice B.

B) 2

8 Why not the others

A is −64.

Wrong: put in x = −3: for x − 3 we use x = 3.

A) −64 ✗ put in x = −3: for x − 3 we use x = 3

C is 5.

Wrong: that is the last number only.

C) 5 ✗ that is the last number only

D is 74.

Wrong: −4 × 9 is −36, not +36.

D) 74 ✗ −4 × 9 is −36, not +36

Let's look back.

First: call it f(x).

Then: make x − 3 zero.

Then: why f(3) is the remainder.

Then: work out f(3).

Then: read the remainder.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer