Factor y^3 - 3y^2 + 2y = 0

Step by step

The question

Question: solve y³ − 3y² + 2y = 0.

The right side is 0 already.

1 Factor completely

Common factor first.

Every term has y.

So we take out y.

So it is y(y² − 3y + 2) = 0.

Careful!

Don't divide both sides by y.

Then the answer y = 0 would be lost.

Now the bracket.

We need two numbers.

They multiply to 2, and they add to −3.

They multiply to a plus number, so the two numbers have the same sign.

They add to a minus number, so both are minus.

We list the pairs that make 2, smallest first.

We add each pair.

We need 3.

1 and 2 give 3.

Yes!

With the signs: −1 and −2.

So it is y(y − 1)(y − 2) = 0.

2 Each factor is 0

Some numbers multiply to 0.

That only happens if one of them is 0.

So we make each factor 0.

Each factor with x gives a small equation.

y = 0 is done already.

In y − 1 = 0, we add 1 on both sides.

In y − 2 = 0, we add 2 on both sides.

So y = 0, y = 1 and y = 2.

Three answers.

3 Check

Let's check: put each answer into y³ − 3y² + 2y = 0.

First y = 0.

Now y = 1.

Now y = 2.

We get 0 every time.

So all three answers are right!

Let's look back.

First: factor completely.

Then: each factor is 0.

Last: check.