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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.
Examples
Example 1 Expand 4(3 x − 2)
4(3 x − 2) = 12 x − 8 Multiply everything inside the bracket by the 4 outside the bracket
Example 2 Expand and simplify 3( x + 5) − 4(2 x + 3)
3( x + 5) − 4(2 x + 3) = 3 x + 15 − 8 x – 12
= 3 − 5 x
1 Expand each set of brackets separately by multiplying ( x + 5) by 3 and (2 x + 3) by − 4 2 Simplify by collecting like terms: 3 x − 8 x = − 5 x and 15 − 12 = 3
Example 3 Expand and simplify ( x + 3)( x + 2)
( x + 3)( x + 2) = x ( x + 2) + 3( x + 2) = x^2 + 2 x + 3 x + 6 = x^2 + 5 x + 6
1 Expand the brackets by multiplying ( x + 2) by x and ( x + 2) by 3
2 Simplify by collecting like terms: 2 x + 3 x = 5 x
Example 4 Expand and simplify ( x − 5)(2 x + 3)
( x − 5)(2 x + 3) = x (2 x + 3) − 5(2 x + 3) = 2 x^2 + 3 x − 10 x − 15 = 2 x^2 − 7 x − 15
1 Expand the brackets by multiplying (2 x + 3) by x and (2 x + 3) by − 5
2 Simplify by collecting like terms: 3 x − 10 x = − 7 x
Practice
1 Expand. a 3(2 x − 1) b −2(5 pq + 4 q^2 ) c −(3 xy − 2 y^2 )
2 Expand and simplify. a 7(3 x + 5) + 6(2 x – 8) b 8(5 p – 2) – 3(4 p + 9) c 9(3 s + 1) –5(6 s – 10) d 2(4 x – 3) – (3 x + 5)
3 Expand. a 3 x (4 x + 8) b 4 k (5 k^2 – 12) c –2 h (6 h^2 + 11 h – 5) d –3 s (4 s^2 – 7 s + 2)
4 Expand and simplify. a 3( y^2 – 8) – 4( y^2 – 5) b 2 x ( x + 5) + 3 x ( x – 7) c 4 p (2 p – 1) – 3 p (5 p – 2) d 3 b (4 b – 3) – b (6 b – 9)
5 Expand 12 (2 y – 8)
6 Expand and simplify. a 13 – 2( m + 7) b 5 p ( p^2 + 6 p ) – 9 p (2 p – 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 21 x^2 – 35 x
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 (2 x + 3)( x – 1) f (3 x – 2)(2 x + 1) g (5 x – 3)(2 x – 5) h (3 x – 2)(7 + 4 x ) i (3 x + 4 y )(5 y + 6 x ) j ( x + 5)^2 k (2 x − 7) 2 l (4 x − 3 y )^2
Extend
9 Expand and simplify ( x + 3)² + ( x − 4)²
10 Expand and simplify.
a x^1 x^2
x x
b
2
x^1
x
Watch out! When multiplying (or dividing) positive and negative numbers, if the signs are the same the answer is ‘+’; if the signs are different the answer is ‘–’.
Surds and rationalising the denominator
A LEVEL LINKS
Scheme of work: 1a. Algebraic expressions – basic algebraic manipulation, indices and surds
Key points
- A surd is the square root of a number that is not a square number, for example 2,^ 3,^ 5,^ etc.
- Surds can be used to give the exact value for an answer.
- (^) ab = a × b
- a^ a b (^) b
- To rationalise the denominator means to remove the surd from the denominator of a fraction.
- To rationalise a b
you multiply the numerator and denominator by the surd b
you multiply the numerator and denominator by (^) b − c
Examples
Example 1 Simplify 50
50 = 25 × 2
= ×
= ×
1 Choose two numbers that are factors of 50. One of the factors must be a square number 2 Use the rule ab = a × b 3 Use 25 = 5
Example 2 Simplify 147 −2 12
= × − ×
= 49 × 3 − 2 4 × 3
= 7 × 3 − 2 × 2 × 3
= 7 3 − 4 3 = 3 3
1 Simplify 147 and 2 12. Choose two numbers that are factors of 147 and two numbers that are factors of
- One of each pair of factors must be a square number 2 Use the rule ab = a × b 3 Use 49 = 7 and 4 = 2
4 Collect like terms
Example 3 Simplify (^) ( 7 + (^2) )( 7 − (^2) )
( 7 +^2 )( 7 −^2 ) = 49 − 7 2 + 2 7 − 4
= 7 – 2 = 5
1 Expand the brackets. A common mistake here is to write (^) ( )
2 7 = 49
2 Collect like terms: 7 2 2 7 7 2 7 2 0
Example 4 Rationalise 1 3
1 3
×
×
1 Multiply the numerator and denominator by 3
2 Use 9 = 3
Example 5 Rationalise and simplify 2 12
2 12
×
× ×
1 Multiply the numerator and denominator by 12
2 Simplify 12 in the numerator. Choose two numbers that are factors of 12. One of the factors must be a square number
3 Use the rule ab = a × b 4 Use 4 = 2
5 Simplify the fraction: 2 12
simplifies to 1 6
4 Rationalise and simplify, if possible.
a^1 5
b^1 11 c^2 7
d^2 8 e^2 2
f^5 5
g^8 24
h^5 45
5 Rationalise and simplify.
a −
b
c −
Extend
6 Expand and simplify (^) ( x + y (^) )( x − y )
7 Rationalise and simplify, if possible.
a
b
x − y
Answers
1 a 3 5 b 5 5
c 4 3 d 5 7 e (^) 10 3 f (^) 2 7 g 6 2 h 9 2
2 a 15 2 b 5
c 3 2 d 3 e 6 7 f (^) 5 3
3 a − 1 b 9 − 3
c 10 5 − 7 d 26 − 4 2
4 a^5 5
b^11 11
c 2 7 7
d^2 2 e 2 f 5
g^3 3 h^1 3
5 a^3 +^5 4
b 2(4^ − 3) 13
c 6(5^ + 2) 23
6 x − y
7 a 3 + 2 2 b
x y
x y
Example 4 Evaluate 4 −^2
2 2
1 Use the rule
m^1
a am
2 Use 42 = 16
Example 5 Simplify
5 2
x
x
5 2
x x
= 3 x^3 6 ÷ 2 = 3 and use the rule
m (^) m n n
a (^) a a
= − to
give
(^5 5 2 ) 2
x (^) x x x
Example 6 Simplify
3 5 4
x x x
×
3 5 3 5 8 4 4 4
x x x x x x x
× +
= x^8 −^4 = x^4
1 Use the rule a m^ × a n^ = am + n
2 Use the rule
m (^) m n n
a (^) a a
=^ −
Example 7 Write 1 3 x
as a single power of x
x x = − Use the rule 1 m am^ a
= − , note that the
fraction 1 3
remains unchanged
Example 8 Write 4 x
as a single power of x
(^12) 1 2
x (^) x
x −
1 Use the rule
1 a n =^ na
2 Use the rule
1 m
am^ a
=^ −
Practice
1 Evaluate. a 14 0 b 3 0 c 5 0 d x^0
2 Evaluate.
a
1 492 b
1 643 c
1 1253 d
1 164
3 Evaluate.
a
3 252 b
5 83 c
3 49 2 d
3 164
4 Evaluate.
a 5 –2^ b 4 –3^ c 2 –5^ d 6 –
5 Simplify.
a
2 3 2
x x x
× (^) b^5 2
x x × x
c
3 3
x x x
× (^) d^3 5
x y x y
e (^) 1 2
y^2 y × y
f
(^12) (^232)
c c × c
g ( ) 23 0
x x
h
(^12 ) 2 3
x x x − x
×
×
6 Evaluate.
a
1 4 −^2 b
2 27 −^3 c
1 9 −^2 × 23
d
1 16 4 × 2 −^3 e
1 9 2 16
^ −
f
2 27 3 64
^ −
7 Write the following as a single power of x.
a^1 x
b^17 x
c^4 x
d^5 x^2 e (^) 3
x
f (^) 3 12 x
Watch out! Remember that any value raised to the power of zero is 1. This is the rule a^0 = 1.
Answers
1 a 1 b 1 c 1 d 1
2 a 7 b 4 c 5 d 2
3 a 125 b 32 c 343 d 8
4 a^1 25
b^1 64
c^1 32
d^1 36
5 a
x (^) b 5 x 2
c 3 x d (^) 2 2
y x
e
1 y^2 f c – g 2 x^6 h x
6 a^1 2
b^1 9
c^8 3 d^1 4
e^4 3
f^16 9
7 a x –1^ b x –7^ c
1 x^4
d
2 x^5 e
1 x −^3 f
2 x −^3
8 a^13 x
b 1 c^5 x
d^5 x^2 e^1 x
f (^) 4 13 x
9 a
1 5 x^2 b 2 x –3^ c^14 3
x −
d
1 2 x −^2 e
1 4 x −^3 f 3 x^0
10 a x^3^ + x −^2 b x^3 + x c x −^2 + x −^7
Factorising expressions
A LEVEL LINKS
Scheme of work: 1b. Quadratic functions – factorising, solving, graphs and the discriminants
Key points
- Factorising an expression is the opposite of expanding the brackets.
- A quadratic expression is in the form ax^2 + bx + c , where a ≠ 0.
- To factorise a quadratic equation find two numbers whose sum is b and whose product is ac.
- An expression in the form x^2 – y^2 is called the difference of two squares. It factorises to ( x – y )( x + y ).
Examples
Example 1 Factorise 15 x^2 y^3 + 9 x^4 y
15 x^2 y^3 + 9 x^4 y = 3 x^2 y (5 y^2 + 3 x^2 ) The highest common factor is 3 x^2 y. So take 3 x^2 y outside the brackets and then divide each term by 3 x^2 y to find the terms in the brackets
Example 2 Factorise 4 x^2 – 25 y^2
4 x^2 – 25 y^2 = (2 x + 5 y )(2 x − 5 y ) This is the difference of two squares as the two terms can be written as (2 x )^2 and (5 y )^2
Example 3 Factorise x^2 + 3 x – 10
b = 3, ac = −
So x^2 + 3 x – 10 = x^2 + 5 x – 2 x – 10
= x ( x + 5) – 2( x + 5)
= ( x + 5)( x – 2)
1 Work out the two factors of ac = −10 which add to give b = 3 (5 and −2) 2 Rewrite the b term (3 x ) using these two factors 3 Factorise the first two terms and the last two terms 4 ( x + 5) is a factor of both terms
Practice
1 Factorise. a 6 x^4 y^3 – 10 x^3 y^4 b 21 a^3 b^5 + 35 a^5 b^2 c 25 x^2 y^2 – 10 x^3 y^2 + 15 x^2 y^3
2 Factorise a x^2 + 7 x + 12 b x^2 + 5 x – 14 c x^2 – 11 x + 30 d x^2 – 5 x – 24 e x^2 – 7 x – 18 f x^2 + x – g x^2 – 3 x – 40 h x^2 + 3 x – 28
3 Factorise a 36 x^2 – 49 y^2 b 4 x^2 – 81 y^2 c 18 a^2 – 200 b^2 c^2
4 Factorise a 2 x^2 + x –3 b 6 x^2 + 17 x + 5 c 2 x^2 + 7 x + 3 d 9 x^2 – 15 x + 4 e 10 x^2 + 21 x + 9 f 12 x^2 – 38 x + 20
5 Simplify the algebraic fractions.
a
2 2
2 x 4 x x x
b
2 2
x x x x
c
2 2
x x x x
d
2 2
x x x
e
2 2
x x x x
f
2 2
x x x x
6 Simplify
a
2 2
x x x
b
2 2
x x x x
c
2 2
x x x
d
2 2
x x x x
Extend
7 Simplify x^^2 +^10 x +^25
8 Simplify
2 2 2
x x x
Hint Take the highest common factor outside the bracket.
Answers
1 a 2 x^3 y^3 (3 x – 5 y ) b 7 a^3 b^2 (3 b^3 + 5 a^2 ) c 5 x^2 y^2 (5 – 2 x + 3 y )
2 a ( x + 3)( x + 4) b ( x + 7)( x – 2) c ( x – 5)( x – 6) d ( x – 8)( x + 3) e ( x – 9)( x + 2) f ( x + 5)( x – 4) g ( x – 8)( x + 5) h ( x + 7)( x – 4)
3 a (6 x – 7 y )(6 x + 7 y ) b (2 x – 9 y )(2 x + 9 y ) c 2(3 a – 10 bc )(3 a + 10 bc )
4 a ( x – 1)(2 x + 3) b (3 x + 1)(2 x + 5) c (2 x + 1)( x + 3) d (3 x – 1)(3 x – 4) e (5 x + 3)(2 x +3) f 2(3 x – 2)(2 x –5)
5 a 2(^ 2) 1
x x
b 1
x x − c x^2 x
x x + e x^3 x
x x −
6 a^3 7
x x
b^2 3 2
x x
c^2 2 3
x x
d^3 4
x x
7 ( x + 5)
8 4(^ 2)
x x
Practice
1 Write the following quadratic expressions in the form ( x + p )^2 + q a x^2 + 4 x + 3 b x^2 – 10 x – 3 c x^2 – 8 x d x^2 + 6 x e x^2 – 2 x + 7 f x^2 + 3 x – 2
2 Write the following quadratic expressions in the form p ( x + q )^2 + r a 2 x^2 – 8 x – 16 b 4 x^2 – 8 x – 16 c 3 x^2 + 12 x – 9 d 2 x^2 + 6 x – 8
3 Complete the square. a 2 x^2 + 3 x + 6 b 3 x^2 – 2 x c 5 x^2 + 3 x d 3 x^2 + 5 x + 3
Extend
4 Write (25 x^2 + 30 x + 12) in the form ( ax + b )^2 + c.
Answers
1 a ( x + 2)^2 – 1 b ( x – 5)^2 – 28
c ( x – 4)^2 – 16 d ( x + 3)^2 – 9
e ( x – 1)^2 + 6 f
(^) x
2 a 2( x – 2)^2 – 24 b 4( x – 1)^2 – 20
c 3( x + 2)^2 – 21 d
(^) x
3 a
(^) x
b
(^) x
c
(^) x
d
(^) x
4 (5 x + 3)^2 + 3
