THE DISTRIBUTIVE LAW
When an equation of the form a (b + c ) is expanded, every term inside the bracket is
multiplied by the number or pronumeral (letter), and the sign that is located outside the
brackets. This rule is known as the Distributive Law.
a (b + c ) = ( a × b) + ( a × c ) = ab + ac
Note: To avoid mistakes, include arrows above or below the terms that are being multiplied.
To expand a quadratic expression we often use “FOIL” i.e. First, Outside, Inside, Last.
(a + b )(c + d ) = ac + ad + bc + bd
When expanding expressions, we remove brackets, and then simplify by collecting like
terms.
There is usually some simplifying to do afterwards which includes collecting like terms.
This process can be extended to expand three binomial factors i.e. cubic expansions.
For example:
Expand (x + 1)( x + 3)(4 − x ) .
Do not attempt to expand all three brackets at one time.
(x + 1)(x + 3)(4 − x )
(
)
= x 2 + 3 x + x + 3 (4 − x )
(
= (4 x
)
= x 2 + 4 x + 3 (4 − x )
2
− x 3 + 16 x − 4 x 2 + 12 − 3 x
* Expand the first two brackets
)
3
= − x + 13 x + 12
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QUESTION 13
Expand the following expressions:
(a)
−5( x + 7)
(b)
(3 x + 1)(2 x − 5) = 3x × 2 x + 3x × −5 + 1 × 2 x + 1 × −5
= 6 x 2 − 15 x + 2 x − 5
= 6 x 2 − 13 x − 5
(c)
2 z ( z − 1) − 3( z + 5)
(d)
5(7 − 2 x )(5 x − 3)
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(e)
(5 x + 7 y ) 2
(f)
( x + 7)( x − 3)( x + 5)
(g)
x x 2 − 3x + 4 − 2 x x 2 − 3x − 5
(
(
)
(
)
)
(
)
= x × x 2 + ( x × −3 x ) + ( x × 4) + − 2 x × x 2 + (− 2 x × −3 x ) + (− 2 x × −5)
= x 3 − 3 x 2 + 4 x − 2 x 3 + 6 x 2 + 10 x
= − x 3 + 3 x 2 + 14 x
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EXPANDING EXPRESSIONS BY RULE
Some quadratic and cubic expressions will always follow a certain pattern, and therefore, we
can expand these expressions by using a standard set of rules.
PERFECT SQUARES
Perfect squares are expressions that are written to the power of two.
For example: x 2 , ( x + 3) 2 and ( x 3 − x + 1) 2 .
Perfect squares may be expanded directly or by applying the following rules:
(a + b )2 = a 2 + 2ab + b 2
(a − b )2 = a 2 − 2ab + b 2
Take care to avoid confusing the following:
(a + b )2 ≠ a 2 + b 2
(a − b )2 ≠ a 2 − b 2
common exam error
QUESTION 14
Expand and simplify each of the following expressions:
(a)
(3a − b) 2 = (3a ) 2 − 2(3a )(b) + (b) 2
= 9a 2 − 6ab + b 2
(b)
(5 x + 7 y ) 2
(c)
(1 + x ) 2
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THE DIFFERENCE OF TWO SQUARES
Given the product of two expressions that consist of the sum and difference of the same
terms, we expand by applying the Difference of Two Squares in reverse.
(a + b )(a − b ) = a 2 − b 2
QUESTION 15
Expand and simplify the following expressions:
(a)
(7 x + 2)(7 x − 2) = (7 x) 2 − (2) 2
= 49 x 2 − 4
(b)
( x − 5 x)( x + 5 x)
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EXPANDING PERFECT CUBES BY RULE
Perfect cubes are expressions that are written to the power of three.
For example: x 3 , ( py ) , ( x + 3) ,
3
3
(ax + 6)3 .
Perfect cubes may be expanded directly or by applying the following rules:
(a + b )3 = a 3 + 3a 2 b + 3ab 2 + b 3
(a − b )3 = a 3 − 3a 2 b + 3ab 2 − b 3
Take care to avoid confusing the following:
(a + b )3 ≠ a 3 + b 3
(a − b )3 ≠ a 3 − b 3
QUESTION 16
Expand the following expressions:
(a)
( 2 x + 3) 3 = ( 2 x ) 3 + 3( 2 x ) 2 (3) + 3( 2 x )(3) 2 + (3) 3
= 8 x 3 + 3( 4 x 2 )(3) + 3( 2 x )(9) + 27
= 8 x 3 + 36 x 2 + 54 x + 27
(b)
(3 x − 8 y ) 3
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QUESTION 17
Expand the following expressions:
3
(a)
 2 x
 4x − 
2

(b)
 2 x 15 
+ 

x
 3
3
QUESTION 18
Expand and simplify the following expressions:
(a)
− 3w(11 + 4 w)
(b)
(17 y − 3)(2 − 5 y )
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(c)
(4w − 3z ) 2
(d)
(3 − 2 x )(3 + 2 x )
(e)
 1
1 − 
x

3
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TECHNIQUES IN FACTORISATION
The process where brackets are inserted into an equation is referred to as factorisation.
Factorisation is the opposite process to expansion.
Expansion
( x + 3)( x − 5) → x 2 − 2 x − 15
←
Factorisation
METHOD
Bring all terms to one side of the equation and simplify by applying one or more of the
following techniques:
(a)
Remove the Highest Common Factor.
(b)
If the polynomial expression consists of TWO terms (binomial expression),
factorise by using one of the following rules:
•
The difference of two squares:
•
The sum or difference of two cubes.
a 2 − b 2 = (a − b )(a + b )
(
)
(
)
a 3 + b 3 = (a + b ) a 2 − ab + b 2
a 3 − b 3 = (a − b ) a 2 + ab + b 2
(c)
If the polynomial expression consists of THREE terms (trinomial expression),
factorise by using one of the following rules:
•
Rules for perfect squares.
a 2 + 2 ab + b 2 = (a + b )
a 2 − 2 ab + b 2 = (a − b )
2
2
•
The FOIL method (to write expressions as linear factors).
•
Completing the Square.
•
Write the equation as a quadratic expression by using substitution.
(Let A = method)
•
The factor theorem and long division.
Note: A quadratic trinomial is an expression of the form: ax 2 + bx + c
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THE DIFFERENCE OF TWO SQUARES
a 2 − b2 =
(
a 2 + b2
)(
)
a 2 − b 2 = ( a + b )( a − b )
The Difference of Two Squares (DOTS) is used to factorise equations that consist of the
difference of two terms (binomial expressions), both of which are perfect squares.
To factorise these expressions:
Step 1: Remove the highest common factor.
Step 2: Take the square root of each entire term.
Step 3: Add and subtract each term.
Note: The sum of two squares ( a 2 + b 2 ) cannot be factorised.
QUESTION 19
Factorise the following expressions:
(a)
x 2 − 49
(b)
9 x 2 − 64 z 2
= ( 3x ) − (8 z )
2
2
= ( 3x − 8 z ) − ( 3x + 8 z )
(c)
15( x − 2) 2 − 135
(d)
16 − 4 x 2 y 6
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QUADRATIC TRINOMIALS
A quadratic trinomial is an expression of the form: ax 2 + bx + c . There are a variety of
methods that may be used to factorise trinomials, including:
•
Rules for perfect squares.
•
Producing two linear factors by trial and error (FOIL).
•
Completing the Square.
FACTORISING QUADRATIC TRINOMIALS BY RULE
If the given equation is a quadratic trinomial (a quadratic equation consisting of three terms),
where one term is two times the product of the square root of the other two terms, then the
equation may be factorised using the following rules:
a 2 + 2ab + b 2 = (a + b )
a 2 − 2ab + b 2 = (a − b )
2
2
Note that the sign in front of the term containing the product (2ab) determines which formula
is to be applied.
QUESTION 20
Factorise x 2 + 6 x + 9 .
Solution
x 2 + 6x + 9
x2
(x
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9
+
3)
2
= ( x + 3)
2
Summer School – Year 11 Mathematics – Book 1
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QUESTION 21
Factorise the following expressions:
(a)
x 2 − 14 x + 49
(b)
4 x 2 + 20 x + 25
(c)
25 z 2 − 60 zy + 36 y 2
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FACTORISING QUADRATIC TRINOMIALS
Many quadratic trinomials (quadratic equations consisting of three terms) can be factorised
to produce two linear factors. This technique involves the construction of two pairs of
brackets, and inserting the appropriate factors by using FOIL in reverse. You should already
be familiar with this process – here are some questions to try, to see if you are already able
to do there!
QUESTION 22
Factorise x 2 + 10 x + 21 .
Solution
Write down the factors of the term involving x 2 :
Factors: x and x
∴ x 2 + 10 x + 21 = ( x
)( x
)
Write down the factors of the term that is independent of x (the constant):
Factors: 21 × 1,
− 21 × −1, 3 × 7, − 3 × −7
Choose the factors in such a way that the sum of the products of the (first term x last
term) and the (outside term x inside term) is equal to the term involving x :
( First
Outside )( Inside
Last )
As the middle term and the last term are positive, both factors of the term that is
independent of x are positive.
∴ x 2 + 10 x + 21 = ( x + 7)( x + 3)
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QUESTION 23
Factorise the following expressions:
(a)
x 2 + 7 x − 18
(b)
x 2 − 13 x + 42
(c)
3 x 2 + 29 x + 18
(d)
9 y 2 + 48 yz + 64 z 2
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