Step by step
The question
Question: expand (2x + 1)³.
Expand means: multiply out all the brackets.
The power 3 means three of the same bracket: (2x + 1) times itself, three times.
1 Multiply the first two
First the first two brackets: (2x + 1)(2x + 1).
The third waits.
Careful!
Cubing a bracket is not cubing each part.
All three brackets must be multiplied out.
Each part of the first bracket multiplies the whole second bracket.
The parts of the first bracket are 2x and 1.
Each one takes the whole second bracket.
What is in front of a bracket multiplies every part inside.
So we write each part times what is in front.
2x times 2x is 4x².
2x times 1 is 2x.
1 times 2x is 2x.
1 times 1 is 1.
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.
2x + 2x is 4x.
It is like 2 apples plus 2 apples: that is 4 apples.
So the expression is 4x² + 4x + 1.
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.
Each part of the first bracket multiplies the whole second bracket.
The parts of the first bracket are 4x² and 4x and 1.
Each one takes the whole second bracket.
What is in front of a bracket multiplies every part inside.
So we write each part times what is in front.
4x² times 2x is 8x³.
4x² times 1 is 4x².
4x times 2x is 8x².
4x times 1 is 4x.
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.
4x² + 8x² is 12x².
It is like 4 apples plus 8 apples: that is 12 apples.
4x + 2x is 6x.
So the expression is 8x³ + 12x² + 6x + 1.
4 Check
Let's check.
We pick a number: x = 2.
We put it into the start: 125.
Now into our answer.
Both give 125.
Same value, so 8x³ + 12x² + 6x + 1 is right!
Let's look back.
First: multiply the first two.
Then: times the third bracket.
Then: group like terms.
Last: check.