Worked examples
The question
Question: expand (x + 1)(x + 2)(x + 3).
Expand (x + 1)(x + 2)(x + 3)
Expand means: multiply out all the brackets.
1 Multiply the first two
First the first two brackets: (x + 1)(x + 2).
The third waits.
= (x + 1)(x + 2)
Careful!
We can't just multiply the first parts and the last parts.
Every part meets every part.
Careful: not x³ + 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 1.
Each one takes the whole second bracket.
= x(x + 2) + 1(x + 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.
= x(x) + x(2) + 1(x) + 1(2)
x times x is x².
x times 2 is 2x.
1 times x is x.
1 times 2 is 2.
= x2 + 2x + x + 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
2x + x is 3x.
It is like 2 apples plus 1 apple: that is 3 apples.
2x + x = 3x
So the expression is x² + 3x + 2.
= x2 + 3x + 2
2 Times the third bracket
Now this answer times the third bracket, (x + 3).
Each part of the long bracket meets every part of the third.
= (x2 + 3x + 2)(x + 3)
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 3x and 2.
Each one takes the whole second bracket.
= x2(x + 3) + 3x(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.
= x2(x) + x2(3) + 3x(x) + 3x(3) + 2(x) + 2(3)
x² times x is x³.
x² times 3 is 3x².
3x times x is 3x².
3x times 3 is 9x.
= x3 + 3x2 + 3x2 + 9x + 2x + 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
3x² + 3x² is 6x².
It is like 3 apples plus 3 apples: that is 6 apples.
9x + 2x is 11x.
3x2 + 3x2 = 6x2
9x + 2x = 11x
So the expression is x³ + 6x² + 11x + 6.
= x3 + 6x2 + 11x + 6
4 Check
Let's check.
We pick a number: x = 2.
We put it into the start: 60.
Start: (2 + 1)(2 + 2)(2 + 3) = 3 × 4 × 5 = 60
Now into our answer.
Both give 60.
Same value, so x³ + 6x² + 11x + 6 is right!
Answer: 2³ + 6(2²) + 11(2) + 6 = 60
Let's look back.
First: multiply the first two.
Then: times the third bracket.
Then: group like terms.
Last: check.
Answer = x3 + 6x2 + 11x + 6
▶ Watch it solved
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 = 2x3 + 3x2 − 11x − 6
▶ Watch it solved
The question
Question: expand (2x − 1)(x + 2)(x − 3).
Expand (2x − 1)(x + 2)(x − 3)
Expand means: multiply out all the brackets.
1 Multiply the first two
First the first two brackets: (2x − 1)(x + 2).
The third waits.
= (2x − 1)(x + 2)
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 2x and −1.
Each one takes the whole second bracket.
= 2x(x + 2) − 1(x + 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.
= 2x(x) + 2x(2) + (−1)(x) + (−1)(2)
− × + = −
Watch the signs.
2x times x is 2x².
2x times 2 is 4x.
(−1) times x is −x.
(−1) times 2 is −2.
= 2x2 + 4x − x − 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
4x − x is 3x.
It is like 4 apples take away 1 apple: that is 3 apples.
4x − x = 3x
So the expression is 2x² + 3x − 2.
= 2x2 + 3x − 2
2 Times the third bracket
Now this answer times the third bracket, (x − 3).
Each part of the long bracket meets every part of the third.
= (2x2 + 3x − 2)(x − 3)
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 2x² and 3x and −2.
Each one takes the whole second bracket.
= 2x2(x − 3) + 3x(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.
= 2x2(x) + 2x2(−3) + 3x(x) + 3x(−3) + (−2)(x) + (−2)(−3)
− × − = +
Watch the signs.
2x² times x is 2x³.
2x² times (−3) is −6x².
3x times x is 3x².
3x times (−3) is −9x.
= 2x3 − 6x2 + 3x2 − 9x − 2x + 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
−6x² + 3x² is −3x².
−9x − 2x is −11x.
−6x2 + 3x2 = −3x2
−9x − 2x = −11x
So the expression is 2x³ − 3x² − 11x + 6.
= 2x3 − 3x2 − 11x + 6
4 Check
Let's check.
We pick a number: x = 2.
We put it into the start: −12.
Start: (2 × 2 − 1)(2 + 2)(2 − 3) = 3 × 4 × (−1) = −12
Now into our answer.
Both give −12.
Same value, so 2x³ − 3x² − 11x + 6 is right!
Answer: 2(2³) − 3(2²) − 11(2) + 6 = −12
Let's look back.
First: multiply the first two.
Then: times the third bracket.
Then: group like terms.
Last: check.
Answer = 2x3 − 3x2 − 11x + 6
▶ Watch it solved
