Expand (x - 2)(x + 3)(2x + 1) Step by Step

Step by step

The question

Question: expand (x − 2)(x + 3)(2x + 1).

Expand (x − 2)(x + 3)(2x + 1)

Expand means: multiply out all the brackets.

1 Multiply the first two

First the first two brackets: (x − 2)(x + 3).

The third waits.

= (x − 2)(x + 3)

Careful!

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

Every part meets every part.

Careful: not 2x³ − 6 ✗

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 + 3) − 2(x + 3)

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(3) + (−2)(x) + (−2)(3)

− × + = −

Watch the signs.

x times x is x².

x times 3 is 3x.

(−2) times x is −2x.

(−2) times 3 is −6.

= x² + 3x − 2x − 6

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

3x − 2x is x.

It is like 3 apples take away 2 apples: that is 1 apple.

3x − 2x = x

So the expression is x² + x − 6.

= x² + x − 6

2 Times the third bracket

Now this answer times the third bracket, (2x + 1).

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

= (x² + x − 6)(2x + 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 x² and x and −6.

Each one takes the whole second bracket.

= x²(2x + 1) + x(2x + 1) − 6(2x + 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.

= x²(2x) + x²(1) + x(2x) + x(1) + (−6)(2x) + (−6)(1)

− × + = −

Watch the signs.

x² times 2x is 2x³.

x² times 1 is x².

x times 2x is 2x².

x times 1 is x.

= 2x³ + x² + 2x² + x − 12x − 6

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 + 2x² is 3x².

It is like 1 apple plus 2 apples: that is 3 apples.

x − 12x is −11x.

x² + 2x² = 3x²

x − 12x = −11x

So the expression is 2x³ + 3x² − 11x − 6.

= 2x³ + 3x² − 11x − 6

4 Check

Let's check.

We pick a number: x = 1.

We put it into the start: −12.

Start: (1 − 2)(1 + 3)(2 × 1 + 1) = (−1) × 4 × 3 = −12

Now into our answer.

Both give −12.

Same value, so 2x³ + 3x² − 11x − 6 is right!

Answer: 2(1³) + 3(1²) − 11(1) − 6 = −12

Let's look back.

First: multiply the first two.

Then: times the third bracket.

Then: group like terms.

Last: check.

Answer