Expand (y + 2)(y - 4)(y + 1) Step by Step

Step by step

The question

Question: expand (y + 2)(y − 4)(y + 1).

Expand (y + 2)(y − 4)(y + 1)

Expand means: multiply out all the brackets.

1 Multiply the first two

First the first two brackets: (y + 2)(y − 4).

The third waits.

= (y + 2)(y − 4)

Careful!

We can't just multiply the first parts and the last parts.

Every part meets every part.

Careful: not y³ − 8 ✗

Each part of the first bracket multiplies the whole second bracket.

(a + b)(c + d) = a(c + d) + b(c + d)

The parts of the first bracket are y and 2.

Each one takes the whole second bracket.

= y(y − 4) + 2(y − 4)

What is in front of a bracket multiplies every part inside.

a(b + c) = ab + ac

So we write each part times what is in front.

= y(y) + y(−4) + 2(y) + 2(−4)

− × + = −

Watch the signs.

y times y is y².

y times (−4) is −4y.

2 times y is 2y.

2 times (−4) is −8.

= y² − 4y + 2y − 8

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the y terms go together.

Each colour is one kind.

like terms = same letters,

same powers

−4y + 2y is −2y.

−4y + 2y = −2y

So the expression is y² − 2y − 8.

= y² − 2y − 8

2 Times the third bracket

Now this answer times the third bracket, (y + 1).

Each part of the long bracket meets every part of the third.

= (y² − 2y − 8)(y + 1)

Each part of the first bracket multiplies the whole second bracket.

(a + b)(c + d) = a(c + d) + b(c + d)

The parts of the first bracket are y² and −2y and −8.

Each one takes the whole second bracket.

= y²(y + 1) − 2y(y + 1) − 8(y + 1)

What is in front of a bracket multiplies every part inside.

a(b + c) = ab + ac

So we write each part times what is in front.

= y²(y) + y²(1) + (−2y)(y) + (−2y)(1) + (−8)(y) + (−8)(1)

− × + = −

Watch the signs.

y² times y is y³.

y² times 1 is y².

(−2y) times y is −2y².

(−2y) times 1 is −2y.

= y³ + y² − 2y² − 2y − 8y − 8

3 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the y² terms go together.

Each colour is one kind.

like terms = same letters,

same powers

y² − 2y² is −y².

It is like 1 apple take away 2 apples: that is 1 apple.

−2y − 8y is −10y.

y² − 2y² = −y²

−2y − 8y = −10y

So the expression is y³ − y² − 10y − 8.

= y³ − y² − 10y − 8

4 Check

Let's check.

We pick a number: y = 2.

We put it into the start: −24.

Start: (2 + 2)(2 − 4)(2 + 1) = 4 × (−2) × 3 = −24

Now into our answer.

Both give −24.

Same value, so y³ − y² − 10y − 8 is right!

Answer: 2³ − 2² − 10(2) − 8 = −24

Let's look back.

First: multiply the first two.

Then: times the third bracket.

Then: group like terms.

Last: check.

Answer