Step by step
The question
Question: expand (x − 1)³.
Expand means: multiply out all the brackets.
The power 3 means three of the same bracket: (x - 1) times itself, three times.
1 Multiply the first two
First the first two brackets: (x − 1)(x − 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 x 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.
Watch the signs.
x times x is x².
x times (−1) is −x.
(−1) times x is −x.
(−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.
−x − x is −2x.
So the expression is x² − 2x + 1.
2 Times the third bracket
Now this answer times the third bracket, (x − 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 x² and −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.
Watch the signs.
x² times x is x³.
x² times (−1) is −x².
(−2x) times x is −2x².
(−2x) times (−1) is 2x.
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.
−x² − 2x² is −3x².
2x + x is 3x.
It is like 2 apples plus 1 apple: that is 3 apples.
So the expression is x³ − 3x² + 3x − 1.
4 Check
Let's check.
We pick a number: x = 2.
We put it into the start: 1.
Now into our answer.
Both give 1.
Same value, so x³ − 3x² + 3x − 1 is right!
Let's look back.
First: multiply the first two.
Then: times the third bracket.
Then: group like terms.
Last: check.