Download Breaking Brackets and Simplifying

ALGEBRA
Like Terms
3a + 4b + a + 5b is an algebraic expression. 3a + a are called like
terms because they both contain the same letter. To simplify
an expression we collect together like terms. You can only add
or subtract when the letters are the same.
Simplifying expressions
Example 1
Example 2
Example 3
Example 4
7b + 3b = 10b
x + 6x + 3x = 10x
5p + 8p – 4p = 9p
m + m – 2m = 0
Collecting together like terms and then simplifying
Example 1
3a + 2b + 4a + 6b – collect together the a’s then the b’s
= 7a + 8b
(3a + 4a + 2b + 6b)
Example 2
8p + 5q – 2p + 7q
= 6p + 12q
(8p – 2p + 5q + 7q)
Example 3
9g – 7h – 4g + 10h
= 5g + 3h
(9g – 4g – 7h + 10h)
Example 4
6c + 12 + 8c
= 14c + 12
(6c + 8c + 12)
Example 5
4v – 2u + 7v + 2u
= 11v
(4v + 7v – 2u + 2u)
Multiplying
When multiplying terms together we can leave out the multiplication
sign. Numbers should be at the front and then letters in alphabetical
order.
Examples
4 x a = 4a
p x 5 = 5p
3 x b x c = 3bc
6e x 4f = 24ef
e x e = e2
c x c x d = c2d
a x b = ab
1 x q = q (don’t need to write in number 1)
7 x h x g = 7gh
8b x 5a = 40ab
3p x 2p = 6p2
4a x 2b x 3c = 24abc
Brackets
4(a + b) is the same as 4 x (a + b). You must multiply everything
inside the bracket by the number outside the bracket.
Example 1
4(a + b)
= 4a + 4b
Example 2
7(p – q)
= 7p – 7q
Example 3
-5(3g + 4)
= -15g - 20
Example 4
6g(2g + 3h)
= 12g2 + 18gh
Example 5
-9u(4u – 5v)
= -36u2 + 45uv
Multiplying out brackets then simplifying
Sometimes you need to multiply out brackets first then simplify.
Example 1
2(5 + x) + 3x
= 10 + 2x + 3x
= 10 + 5x
Example 2
4(3y - 2) – 5
= 12y - 8 – 5
= 12y - 13
Example 3
5(2a + 7) – 9a
= 10a + 35 – 9a
= a + 35
Example 4
7 + 5(a - 3)
= 7 + 5a - 15
= 5a - 8
DON’T add 7 + 5
Sometimes you need to multiply out two brackets then simplify.
Example 1
2(3n + 4) + 3(3n + 5)
= 6n + 8 + 9n + 15
= 15n + 23
Example 2
4(2p + 6q) - 3(p – 4q)
= 8p + 24q - 3p + 12q
= 5p + 36q
Pairs of brackets F – First
(F
O – Outside
(O
I – Inside
(
L – Last
(
FOIL
) (F
) (
I) (I
L) (
)
O)
)
L)
Example1
(x + 3) (x + 5)
= x2 + 5x + 3x + 15
= x2 + 8x + 15
Example 2
(p + 2) (p - 8)
= p2 – 8p + 2p - 16
= p2 – 6p – 16
Example 3
(2x - 3) (5x - 1)
= 10x2 – 2x – 15x + 3
= 10x2 – 17x + 3
Example 4
(3d - 2) (4d + 5)
= 12d2 + 15d – 8d - 10
= 12d2 + 7d - 10
Squaring brackets
To square brackets we write the two brackets side by side and then
multiply out as before.
Example 1
(x +y) 2
= (x + y) (x + y)
= x2 +xy + xy + y2
= x2 + 2xy + y2
Example 2
(3m – 5)2
= (3m – 5) (3m – 5)
= 9m2 - 15m – 15m + 25
= 9m2 – 30m + 25
Other brackets
If one of the brackets contains more then two terms we cannot use
FOIL.
Take each term in the first bracket and multiply the second term by
it.
Example 1
(m + 2) (m2 – 4m + 1)
= m (m2 – 4m + 1) + 2(m2 – 4m + 1)
= m3 – 4m2 + m + 2m2 – 8m + 2
= m3 – 2m2 – 7m +2
Example 2
(p - 3) (p2 + 5p - 2)
= p (p2 + 5p - 2) - 3(p2 + 5p - 2)
= p3 + 5p2 – 2p - 3p2 – 15p + 6
= p3 + 2p2 - 17p + 6
(watch the signs!)