Factor completely 4y^3 - 36y Step by Step

Step by step

The question

Question: factor completely 4y³ − 36y.

Completely means: until no part can be factored any more.

4y³ − 36y

1 Common factor first

All the numbers can be divided by 4.

Every term has y.

So the common factor is 4y.

4, 36: all in the 4 times table

every term has y

We divide every term by it.

What is left goes in the bracket.

4y³ ÷ 4y = y²

−36y ÷ 4y = −9

So it is 4y(y² − 9).

= 4y(y² − 9)

2 Count the terms

The bracket has 2 terms.

With 2 terms we look for two squares or two cubes.

y² − 9 is one square take away another square.

y² is y squared.

9 is 3 squared.

A square minus a square is the plus bracket times the minus bracket.

a² − b² = (a + b)(a − b)

y² = (y)², 9 = 3²

So we get 4y(y + 3)(y − 3).

= 4y(y + 3)(y − 3)

3 Check each bracket

Brackets like y + 3 have just y and a number.

They can't be factored.

y + 3, y − 3: just y and a number

No bracket can be factored any more.

So we are done.

Careful!

Don't forget the common factor 4y in the answer.

Careful: not (y² − 9) ✗

4 Check

Let's check: we multiply it back out.

Each part of one bracket times each part of the other.

Check: (y + 3)(y − 3) = y² − 3y + 3y − 9

= y² − 9

Then the common factor times every part.

4y(y² − 9) = 4y³ − 36y

We get 4y³ − 36y again.

So it is right!

Let's look back.

First: common factor first.

Then: count the terms.

Then: check each bracket.

Last: check.

Answer