Expanding and simplifying algebraic expressions

Contributed by:

NEO Tue, Mar 15, 2022 01:09 PM UTC

This pdf includes the following topics:-
Expanding brackets
Simplifying expression
Key points
Examples
Practice Problems

1. Expanding brackets
and simplifying expressions
A LEVEL LINKS
Scheme of work: 1a. Algebraic expressions – basic algebraic manipulation, indices and surds
Key points
 When you expand one set of brackets you must multiply everything inside the bracket by
what is outside.
 When you expand two linear expressions, each with two terms of the form ax + b, where
a ≠ 0 and b ≠ 0, you create four terms. Two of these can usually be simplified by collecting
like terms.
Example 1 Expand 4(3x − 2)
4(3x − 2) = 12x − 8 Multiply everything inside the bracket
by the 4 outside the bracket
Example 2 Expand and simplify 3(x + 5) − 4(2x + 3)
3(x + 5) − 4(2x + 3) 1 Expand each set of brackets
= 3x + 15 − 8x – 12 separately by multiplying (x + 5) by
3 and (2x + 3) by −4
= 3 − 5x 2 Simplify by collecting like terms:
3x − 8x = −5x and 15 − 12 = 3
Example 3 Expand and simplify (x + 3)(x + 2)
(x + 3)(x + 2) 1 Expand the brackets by multiplying
= x(x + 2) + 3(x + 2) (x + 2) by x and (x + 2) by 3
= x2 + 2x + 3x + 6
= x2 + 5x + 6 2 Simplify by collecting like terms:
2x + 3x = 5x
Example 4 Expand and simplify (x − 5)(2x + 3)
(x − 5)(2x + 3) 1 Expand the brackets by multiplying
= x(2x + 3) − 5(2x + 3) (2x + 3) by x and (2x + 3) by −5
= 2x2 + 3x − 10x − 15
= 2x2 − 7x − 15 2 Simplify by collecting like terms:
3x − 10x = −7x

2. 1 Expand. Watch out!
a 3(2x − 1) b −2(5pq + 4q2)
When multiplying (or
c −(3xy − 2y2)
dividing) positive and
2 Expand and simplify. negative numbers, if
the signs are the same
a 7(3x + 5) + 6(2x – 8) b 8(5p – 2) – 3(4p + 9)
the answer is ‘+’; if the
c 9(3s + 1) –5(6s – 10) d 2(4x – 3) – (3x + 5) signs are different the
answer is ‘–’.
3 Expand.
a 3x(4x + 8) b 4k(5k2 – 12)
c –2h(6h2 + 11h – 5) d –3s(4s2 – 7s + 2)
4 Expand and simplify.
a 3(y2 – 8) – 4(y2 – 5) b 2x(x + 5) + 3x(x – 7)
c 4p(2p – 1) – 3p(5p – 2) d 3b(4b – 3) – b(6b – 9)
5 Expand 12 (2y – 8)
6 Expand and simplify.
a 13 – 2(m + 7) b 5p(p2 + 6p) – 9p(2p – 3)
7 The diagram shows a rectangle.
Write down an expression, in terms of x, for the area of
the rectangle.
Show that the area of the rectangle can be written as
21x2 – 35x
8 Expand and simplify.
a (x + 4)(x + 5) b (x + 7)(x + 3)
c (x + 7)(x – 2) d (x + 5)(x – 5)
e (2x + 3)(x – 1) f (3x – 2)(2x + 1)
g (5x – 3)(2x – 5) h (3x – 2)(7 + 4x)
i (3x + 4y)(5y + 6x) j (x + 5)2
k (2x − 7)2 l (4x − 3y)2
9 Expand and simplify (x + 3)² + (x − 4)²
10 Expand and simplify.
2
 1  2  1
a  x   x   b x 
 x  x  x

3. 1 a 6x – 3 b –10pq – 8q2
c –3xy + 2y2
2 a 21x + 35 + 12x – 48 = 33x – 13
b 40p – 16 – 12p – 27 = 28p – 43
c 27s + 9 – 30s + 50 = –3s + 59 = 59 – 3s
d 8x – 6 – 3x – 5 = 5x – 11
3 a 12x2 + 24x b 20k3 – 48k
c 10h – 12h3 – 22h2 d 21s2 – 21s3 – 6s
4 a –y2 – 4 b 5x2 – 11x
c 2p – 7p2 d 6b2
5 y–4
6 a –1 – 2m b 5p3 + 12p2 + 27p
7 7x(3x – 5) = 21x2 – 35x
8 a x2 + 9x + 20 b x2 + 10x + 21
c x2 + 5x – 14 d x2 – 25
e 2x2 + x – 3 f 6x2 – x – 2
g 10x2 – 31x + 15 h 12x2 + 13x – 14
i 18x2 + 39xy + 20y2 j x2 + 10x + 25
k 4x2 − 28x + 49 l 16x2 − 24xy + 9y2
9 2x2 − 2x + 25
2 1
10 a x2 1  b x2  2 
x2 x2

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