Expand and simplify (x + 2)(x − 1)(x + 4). Step by Step

Step by step

The question

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

Expand (x + 2)(x − 1)(x + 4)

Expand means: multiply out all the brackets.

1 Multiply the first two

First the first two brackets: (x + 2)(x − 1).

The third waits.

(x + 2)(x − 1)

Careful!

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

Every part meets every part.

Careful: not x³ − 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 x and 2.

Each one takes the whole second bracket.

= x(x − 1) + 2(x − 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.

= x(x) + x(−1) + 2(x) + 2(−1)

Plus times minus is minus.

+ × − = −

x times x is x²: x squared.

x × x = x²

x times (−1) is −x.

2 times x is 2x.

2 times (−1) is −2.

= x² − x + 2x − 2

Terms are the parts between the + and − signs.

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

All the x terms go together.

Each colour is one kind.

like terms = same letters,

same powers

x alone means 1x.

−x + 2x is x.

1x is just written x.

x = 1x

−x + 2x = x

So the expression is x² + x − 2.

= x² + x − 2

2 Times the third bracket

Now this answer times the third bracket, (x + 4).

(x² + x − 2)(x + 4)

Each part of the long bracket multiplies the whole third bracket.

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

The parts of the long bracket are x² and x and −2.

Each one takes the whole third bracket.

= x²(x + 4) + x(x + 4) − 2(x + 4)

The minus sign in front of the 2 goes with it.

So we multiply by −2.

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.

= x²(x) + x²(4) + x(x) + x(4) + (−2)(x) + (−2)(4)

Minus times plus is minus.

− × + = −

x times x is x²: x squared.

x × x = x²

x² times x is x³.

x² times 4 is 4x².

x times 4 is 4x.

(−2) times x is −2x.

(−2) times 4 is −8.

= x³ + 4x² + x² + 4x − 2x − 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 x² terms go together.

Each colour is one kind.

like terms = same letters,

same powers

x² alone means 1x².

4x² + x² is 5x².

4x − 2x is 2x.

x² = 1x²

4x² + x² = 5x²

4x − 2x = 2x

So the expression is x³ + 5x² + 2x − 8.

= x³ + 5x² + 2x − 8

4 Check

Let's check.

We pick a number: x = 2.

We put it into the start.

The brackets give 4, 1, 6.

4 times 1 times 6 is 24.

Start: (2 + 2)(2 − 1)(2 + 4) = 4 × 1 × 6 = 24

Now into our answer.

The parts are 8, 20, 4, −8.

8 plus 20 plus 4 minus 8 is 24.

Both give 24.

Same value, so x³ + 5x² + 2x − 8 is right!

Answer: 2³ + 5(2²) + 2(2) − 8 = 8 + 20 + 4 − 8 = 24

Let's look back.

First: multiply the first two.

Then: times the third bracket.

Then: group like terms.

Last: check.

Answer