Download Topic 1: Algebra

Topic 1: Algebra
You should already be familiar with adding and subtracting algebraic terms.
Multiplying terms
First multiply the numbers, then multiply the letters, try to remember to write the letters in
alphabetical order.
e.g.
3e x 5d = 15de
4a²b x 3abc² = 12a3b²c²
Dividing terms
First divide the numbers, then divide the letters. Dividing letters is sometimes called cancelling.
6ab
e.g. 8a ÷ 2 = 4a
 3b
2a
6y ÷ 2y = 3
4 p 2q
p
9x ÷ x = 9

2
12 pq
3q
Another name for a power is an index
25
The plural of index is indices.
Power or
index
Multiplying algebraic expressions with powers
When you multiply powers of the same letter you simply add the indices.
e.g. x3 X x8 = x3 + 8
= x11
2x2 X 5x7 = 10x2+7
= 10x9
x3y2 X xy4 = x3+1y2+4
= x4y6
(x3y²)² = x3y² X x3y²
= x3+3y2+2
= x6y4
(3x4)² = 3x4 x 3x4
= 9x4+4
= 9x8
Dividing Algebraic expressions with powers
When you divide powers of the same letter you simply subtract the indices.
e.g. x8 ÷ x5 = x8-5
= x3
6y6 ÷ 2y4 = 3y6-4
= 3y2
Expanding brackets
If an algebraic expression contains a bracket, the bracket can be removed or expanded by multiplying
everything inside the bracket by the number or letter immediately outside the bracket.
e.g. 2(x + 3) = 2 X x + 2 X 3
= 2x + 6
3(4a + 5) = 3 X 4a + 3 X 5
= 12a +15
x(x – 5) = x X x – 5 X x
= x² - 5x
Always remove brackets before attempting to tidy up an algebraic expression and take care with
signs.
e.g. 2m – 3m(m – 4) = 2m -3m X m - -3m X 4
= 2m -6m²+ 12m
= 14m – 6m²
Factorising
Factorising is the opposite operation to removing brackets.
To do this we look for a factor that is common to everything in the algebraic expression and this goes
immediately outside the bracket, appropriate terms are put inside the bracket so that if the bracket
was expanded the original expression would be obtained.
e.g.
4x – 6 = 2(2x – 3)
2ab² - 6b3 = 2b²(a – 3b)
2x² - 6x = 2x(x – 3)
Solving simple equations
e.g. x + 3 = 7
x=7–3
x=4
2x + 8 = 28
2x = 28 – 8
2x = 20
x = 20 ÷ 2
x = 10
¼x + 1 = 21
¼x = 21 – 1
¼x = 20
x – 4 =9
x=9+4
x = 13
5x = 30
x = 30 ÷ 5
x=6
3x – 7 = 29
3x = 29 + 7
3x = 36
x = 36 ÷ 3
x = 12
x/4 = 6
x=6X4
x = 24
9/x = 27
9 = 27 X x
9 = 27x
10/x -2 = 5
10/x = 5 + 2
10/x = 7
x = 20 X 4
x = 80
9 ÷ 27 = x
1/3 = x
10 = 7x
10 ÷ 7 = x
1.429 = x
5 – 3x = -1
- 3x = -1 -5
- 3x = -6
x = -6 ÷ -3
x=2
Equations with brackets
Expand the brackets first and then solve the equation as normal.
e.g. 4(5y – 6) = 16
20y – 24 = 16
20y = 16 +24
20y = 40
Y = 40 ÷ 20
Y=2
Equations with letters on both sides
When there are letters on both sides of an equation, rearrange the equation so there are only letters
on one side and only numbers on the other side.
e.g.
5x – 8 = 3x + 4
6(x-3) + 2(x+5) = 3(2x-1)
5x – 3x = 4 + 8
6x – 18 + 2x + 10 = 6x -3
2x = 12
8x -8 = 6x - 3
x = 12 ÷ 2
8x -6x = -3 + 8
x=6
2x = 5
x=5÷2
x = 2.5
Multiplying one bracket by another bracket
Everything inside the second bracket needs to be multiplied by everything inside the first bracket.
e.g (x-2)(x + 4)
(3x + 1)(2x – 5)
(3x – 4)² = (3x -4)(3x -4)
X x
-2
x
x² -2x
-8
+4 +4x
= x² - 2x + 4x -8
= x² + 2x -8
X 3x +1
2x 6x² +2x
-5 -15x -5
= 6x² + 2x – 15x – 5
= 6x² - 13x -5
X 3x -4
3x 9x² -12x
-4 -12x +16
= 9x² -12x -12x + 16
= 9x² - 24x +16