Academic Skills Advice
Algebra Refresher Sheet 1
Writing Algebraic Expressions:
Writing an algebraic expression is like writing a sentence in maths instead of English. You
do this by assigning letters to numbers. An algebraic expression is a set of instructions on
how to perform a calculation.
Examples:

Write the following as an algebraic expression:
Five times a number minus three times another number.
First I need to assign letters to the ‘unknown’ numbers. I will call the first one ‘n’ and the
second one ‘m’ so now I have:
Five times n minus three times m.
(Notice I have replaced the 1st and
2nd number with ‘n’ and ‘m’).
Next I replace the words with maths symbols so that I have:
5xn–3xm
Tidy up:
5n – 3m
This is our expression.
𝒙

A number divided by three: can be written as

Half of a number plus quarter of another number: can be written
𝟑
𝒙
𝟐
+
𝒚
𝟒
You can choose any letter but make sure that if you want it to mean a different number you
choose a different letter.
Simplifying Expressions:
Once you have an algebraic expression it can be simplified by collecting all the ‘like terms’
together (i.e. combining things that are the same letter or combination of letters). If an
expression includes brackets then you may need to multiply out the brackets first to see
what will combine.
Examples:
 Simplify the following expression:
3a + 7b – 2c – 4b – 6c + a
Collect together any letters that are the same: 3a + a = 4a
7b – 4b = 3b
-2c – 6c = -8c
So we have:
4a + 3b – 8c
© H Jackson 2008 / Academic Skills
1
 Simplify the following: 3a (a – b) + 5ab
First remove the bracket to see what you can combine. To do this you need to multiply the
bit outside the bracket by every bit (term) inside the bracket.
3a x a = 3a2
3a x –b = - 3ab
So after multiplying out the brackets we have: 3a2 - 3ab + 5ab = 3a2 + 2ab
Factorisation:
An expression can be simplified further by factorising it. This means taking out any
common factors (i.e. numbers or letters that are in every term of the expression). This
involves putting the brackets back in.
Examples:

Simplify the following: 8𝑥 + 3𝑥𝑦 + 9𝑥𝑦 − 4𝑥
Collect like terms: 8𝑥 − 4𝑥 = 4𝑥
3𝑥𝑦 + 9𝑥𝑦 = 12𝑥𝑦
4𝑥 + 12𝑥𝑦
So we have:
Now we can factorise because each term will divide by 4 and 𝑥. (This is the opposite of
multiplying out the bracket.)
4𝑥
4𝑥
12𝑥𝑦
=1
4𝑥
= 3𝑦
Put the bit you have divided by outside the bracket.
4𝑥(1 + 3𝑦)
So now we have:
 Simplify the following expression:
3ab2 + 4a2b – b2a – 5ab – 6a2b
This looks more complicated but remember you can collect any terms that are the same.
Notice that ab2 is the same as b2a (just written the other way round) but is not the same as
a2b.
Collect like terms: 3ab2 – b2a = 2ab2
4a2b – 6a2b = -2a2b
-5ab (there is no other ab term)
So we have:
2ab2 - 2a2b - 5ab
Now we have an expression that we can factorise as each term includes an ‘a’ and a ‘b’ so
we can divide each term by ab.
2𝑎𝑏 2
𝑎𝑏
−2𝑎2 𝑏
= 2𝑏
So we have
𝑎𝑏
= −2𝑎
ab(2b -2a -5)
© H Jackson 2008 / Academic Skills
−5𝑎𝑏
𝑎𝑏
= −5
(this looks tidier and simpler)
2
Practice Questions:
Write algebraic expressions for the following:
1)
2)
3)
4)
5)
Four times a number plus seven.
Six times a number take away five times another number.
Three times a number plus a number divided by four take away seven.
A number plus two all multiplied by three.
Three times a number plus four all divided by three.
Simplify the following:
1)
3d + 7d
2)
8e – 5e
3)
3d + 4e – 2d + 6e
4)
d2 + 3d – e2 + 4d
5)
-4 × 3d
6)
12d2ef3  4de2f
7)
9e2d  27e2d2
8)
8(𝑥 + 2𝑦) + 3(2𝑥 − 3𝑦)
9)
3(2𝑥 + 2𝑦) + 4(𝑥 − 2𝑦) − 2(𝑥 − 3𝑦)
10) 15𝑥𝑦 + 4𝑦 − 5𝑦𝑥 + 6𝑦
11) 6a + 6b
12) 3b – 3b2
13) 4𝑥 2 + 16𝑥
14) 27c3 – 9c
15) 12mn – 4nm +12m
© H Jackson 2008 / Academic Skills
3