Worked examples
The question
Question: simplify 3(2x − 5) + 4x.
Simplify means: write it shorter, with the same value.
3(2x − 5) + 4x
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Open the brackets
The 3 in front of the bracket multiplies every part inside.
a(b + c) = ab + ac
So we write 3 times each part.
= 3(2x) + 3(−5) + 4x
− × + = −
Watch the signs.
3 times 2x is 6x.
3 times (−5) is −15.
= 6x − 15 + 4x
2 Group like terms
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
We move like terms next to each other.
Each term takes its sign with it.
= 6x + 4x − 15
6x + 4x is 10x.
It is like 6 apples plus 4 apples: that is 10 apples.
6x + 4x = 10x
So the expression is 10x − 15.
= 10x − 15
3 Check
Let's check.
We pick a small number for each letter: x = 2.
Any number works, and a small one keeps the sums easy.
We put it into the question.
Question: 3 × (2 × 2 − 5) + 4 × 2
= 3 × (−1) + 8 = −3 + 8 = 5
Now into our answer.
Answer: 10 × 2 − 15 = 20 − 15 = 5
Both give 5.
Same value, so 10x − 15 is right!
Both give 5
4 Pick the choice
Our answer is 10x − 15.
That is choice C.
C) 10x − 15
5 Why not the others
A is 2x − 15.
Wrong: 6x − 15 − 4x: took 4x away.
A) 2x − 15 ✗ 6x − 15 − 4x: took 4x away
B is 6x − 11.
Wrong: 6x − 15 + 4: added 4 to the number.
B) 6x − 11 ✗ 6x − 15 + 4: added 4 to the number
D is 10x − 5.
Wrong: 3 × (−5) is −15, not −5.
D) 10x − 5 ✗ 3 × (−5) is −15, not −5
Let's look back.
First: open the brackets.
Then: group like terms.
Then: check.
Then: pick the choice.
Last: why not the others.
▶ Watch it solved
The question
Question: simplify (x + 4)(x − 4).
Simplify means: write it shorter, with the same value.
(x + 4)(x − 4)
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Split the first bracket
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 4.
Each one takes the whole second bracket.
= x(x − 4) + 4(x − 4)
2 Open the brackets
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(−4) + 4(x) + 4(−4)
− × + = −
x times x is x²: x squared.
x × x = x²
Watch the signs.
x times x is x².
x times (−4) is −4x.
4 times x is 4x.
4 times (−4) is −16.
= x2 − 4x + 4x − 16
3 Group like terms
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 + 4x is 0.
−4x + 4x = 0
So the expression is x² − 16.
= x2 − 16
4 Check
Let's check.
We pick a small number for each letter: x = 2.
Any number works, and a small one keeps the sums easy.
We put it into the question.
Question: (2 + 4)(2 − 4) = (6)(−2) = −12
Now into our answer.
Answer: 22 − 16 = 4 − 16 = −12
Both give −12.
Same value, so x² − 16 is right!
Both give −12
5 Pick the choice
Our answer is x² − 16.
That is choice C.
C) x2 − 16
6 Why not the others
A is x² + 16.
Wrong: 4 × (−4) is −16, not +16.
A) x2 + 16 ✗ 4 × (−4) is −16, not +16
B is x² − 8.
Wrong: 4 × 4 is 16, not 8.
B) x2 − 8 ✗ 4 × 4 is 16, not 8
D is x² − 8x − 16.
Wrong: −4x + 4x is 0, not −8x.
D) x2 − 8x − 16 ✗ −4x + 4x is 0, not −8x
Let's look back.
First: split the first bracket.
Then: open the brackets.
Then: group like terms.
Then: check.
Then: pick the choice.
Last: why not the others.
▶ Watch it solved
The question
Question: factor x² − 2x − 15.
Factor means: write it as a times, with brackets.
x2 − 2x − 15
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find two numbers
First, see how two brackets open.
(x + 2)(x + 3) gives x² + 3x + 2x + 6.
That is x² + 5x + 6.
Look: (x + 2)(x + 3)
= x2 + 3x + 2x + 6
= x2 + 5x + 6
2 and 3 multiply to make the last number, 6.
And they add to make the middle number, 5.
2 × 3 = 6 → last number
2 + 3 = 5 → middle number
Factoring goes backwards.
We need two numbers.
They multiply to −15, and they add to −2.
? × ? = −15
? + ? = −2
They multiply to a minus number, so one number is plus and one is minus.
Only plus times minus gives a minus.
They add to a minus number, so the bigger one is minus.
When we add a plus and a minus, the bigger one wins.
× is − → one +, one −; bigger one −
We list the pairs that make 15, smallest first.
One is minus, so adding them is a take away.
We take the smaller from the bigger.
Without the signs, we need 2.
3 and 5 give 2.
Yes!
With the signs: 3 and −5.
1 × 15: 15 − 1 = 14
3 × 5: 5 − 3 = 2 ✓
The two numbers go in the brackets.
So we get (x + 3)(x − 5).
= (x + 3)(x − 5)
2 Check
Let's check: we multiply it back out.
Each part of the first bracket times each part of the second.
Check: (x + 3)(x − 5) = x2 − 5x + 3x − 15
= x2 − 2x − 15
We get x² − 2x − 15 again.
So (x + 3)(x − 5) is right!
3 Pick the choice
Our answer is (x + 3)(x − 5).
That is choice C.
C) (x − 5)(x + 3)
4 Why not the others
A is (x + 5)(x − 3).
Wrong: 5 + (−3) = +2, not −2.
A) (x + 5)(x − 3) ✗ 5 + (−3) = +2, not −2
B is (x − 15)(x + 1).
Wrong: −15 + 1 = −14, not −2.
B) (x − 15)(x + 1) ✗ −15 + 1 = −14, not −2
D is (x − 5)(x − 3).
Wrong: (−5) × (−3) = +15, not −15.
D) (x − 5)(x − 3) ✗ (−5) × (−3) = +15, not −15
Let's look back.
First: find two numbers.
Then: check.
Then: pick the choice.
Last: why not the others.
▶ Watch it solved
