GCSE Factorising and Simplifying Algebraic

Report
GCSE: Quadratic Functions and
Simplifying Rational Expressions
Dr J Frost ([email protected])
Last modified: 25th August 2013
Factorising Overview
Factorising means :
To turn an expression into a product of factors.
Year 8 Factorisation
2x2 + 4xz
Factorise
So what factors can we
see here?
2x(x+2z)
Year 9 Factorisation
x2 + 3x + 2
Factorise
(x+1)(x+2)
A Level Factorisation
2x3 + 3x2 – 11x – 6
Factorise
(2x+1)(x-2)(x+3)
Factor Challenge
5 + 10x
x – 2xz
x2y – xy2
10xyz – 15x2y
xyz – 2x2yz2
+ x2y2
Exercises
Extension Question:
What integer (whole number) solutions
are there to the equation xy + 3x = 15
1) 2 − 4 = 2  −? 2
2)  +  = ( +? 1)
Answer:   + 3 = 15. So the two
3)  − 2 =   −? 2
expressions we’re multiplying can be 1 ×
15, 15 × 1, 3 × 5, 5 × 3, −1 × −15, …
4) 6 − 3 = 3(2 ?− )
? ) of
This gives solutions (,
5)  +  = ( ?+ 1)
,  , ,  , ,  , , − ,
−, − , −, − , −, − , (−, −)
6)  2  + 2 =   2 ?+ 2
7)  3  +  2 =  ?2 + 
8) 5 + 10 = 5  ?+ 2
9) 12 2 − 8 2  = 4 2 ?− 2
? +3
10) 553 + 332 = 112 5
11) 64 + 83 + 102 = 22 (32 +
? 4 + 5)
12) 10 3  2 + 5 2  3 + 15 2  2 = 5 2  2 (2?+  + 3)
Factorising out an expression
It’s fine to factorise out an entire expression:
 +2 −3 +2
→
? − 3)
( + 2)(
2
 +1 +2 +1
→
2 +  + 2 ?  + 1
 2 + 1 +  2 + 1
?
→ ( + )(2
+ 1)
2 2 − 3 2 +  2 − 3
→ (5 − 6)(2 ?− 3)
Harder Factorisation
 +  −  − 
? − )
= ( − )(
 +  +  + 1
?
= ( + 1)(
+ 1)
Exercises
Edexcel GCSE Mathematics Textbook
Page 111 – Exercise 8D
Q1 (right column), Q2 (right column)
Expanding two brackets
1
2
3
4
5
6
7
8
9
10
11
12
13
14
 + 1  + 2 =  2 + 3?+ 2
?2
 + 2  − 1 = 2 +  −
 − 3  − 4 =  2 − 7?+ 12
 + 1 2 =  2 + 2?+ 1
 − 5 2 =  2 − 10
? + 25
 − 10 2 =  2 − 20
? + 100
 +   + 2 =  2 + 3? + 2 2
? − 2
 +  3 −  = 32 + 2
? + 2 2
3 − 2  −  = 3 2 − 5
 +   +  =  +  +
?  + 
2 + 3 3 − 4 = 6 2 + ?− 12 2
 −   +  = 2 −  2 ?
 +  2 =  2 + 2
? + 2
2 − 3 2 = 4 2 − 12
? + 9 2
Faster expansion of squared brackets
There’s a quick way to expand squared brackets involving two terms:
+
2
=  2 + 2
? + 2
2 − 
2
? + 2
= 4 2 − 4
3 + 4
2
= 92  2 + 24
? + 16 2
7 − 2
2
= 49 2  2 − 28
+ 4 2
?
Four different types of factorisation
1. Factoring out a term
2 2 + 4 = 2  +? 2
2.  +  + 
 2 + 4 − 5 =  + 5 ?  − 1
3. Difference of two squares
4.  +  + 
4 2 − 1 = 2 + 1 ?2 − 1
? − 1)
2 2 +  − 3 = (2 + 3)(
Strategy: either split the middle
term, or ‘go commando’.
2.  2 +  + 
Which is ( + )( + )?
How does this suggest we can factorise say  2 + 3 + 2?
 2 −  − 30 =  + 5 ?  − 6
Is there a good strategy for working out which
numbers to use?
2.  2 +  + 
1
2
3

4
5
6
7
8
9
10
 2 + 4 + 3 =  + 3 ?  + 1
 2 − 8 + 7 =  − 1 ?  − 7
 2 + 2 − 8 =  + 4 ?  − 2
 2 + 16 − 36 =  + 18?  − 2
 2 −  − 56 = ( + 7)(
? − 8)
 2 + 3 − 54 =  + 9 ? − 6
 2 − 3 − 54 =  − 6 ? + 9
 2 + 15 + 54 =  + 6 ? + 9
 2 + 4 + 4 =  + 2 2?
 2 − 14 + 49 =  − 7 2?
 4 + 5 2 + 4 =  2 + 1 ? 2 + 4
3. Difference of two squares
Firstly, what is the square root of:
4 2 = 2 ?
25 2 = 5 ?
16 2  2 = 4?
 4  4 =  2 ?2
9 −6
2
? 6)
= 3( −
3. Difference of two squares
3
3
2
2
2
4 − 9
=(
+
)(
Click to Start
Bromanimation
−
)
3. Difference of two squares
2
? − )
1 −  = (1 + )(1
+1
2
49 − 1 − 
(In your head!)
− −1
2
2
= 4
?
? + )
= (8 − )(6
512 − 492 = 200?
18 2 − 50 2 = 2 3 + 5 ? 3 − 5
2 + 1
2
−9 −6
2
= 5 − 17 ?− + 19
3. Difference of two squares
Exercises:
1
2
3
4
5
6
7
8
9
10
42 − 1 = 2 + 1 ? 2 − 1
4 −  2 = (2 + )(2
? − )
144 −  2 = 12 +  ? 12 − 
 + 1 2 − 25 =  + 6 ? − 4
?
7.642 − 2.362 = 52.8
22 − 32 = 2  + 4 ?  − 4
3 2 − 75 2 = 3  + 5?  − 5
42 − 64 2 = 4  + 4 ?  − 4
9  + 1 2 − 42 = 5 + 3 ? + 3
50 2 + 1 2 − 18 1 −  2 = 2(7 + 8)(13
+ 2)
?
4.  2 +  + 
2
2
?
+  − 3 = (2 + 3)(
− 1)
Factorise using:
a. The ‘commando’ method*
b. Splitting the middle term
* Not official mathematical terminology.
4.  2 +  + 
2 2 + 11 + 12 =  + 4 2 ?+ 3
6 2 − 7 − 3 = (2 − 3)(3? + 1)
? 3
2 2 − 5 + 3 2 =  −  2 −
6 2 − 3 − 3 = 3( − 1)(2? + 1)
Exercises
1
2
3
4
5
6
7
8
9
10
11
N
N
? + 1)
22 + 3 + 1 = (2 + 1)(
32 + 8 + 4 = (3 + 2)(
? + 2)
22 − 3 − 9 = (2 + 3)(
? − 3)
42 − 9 + 2 = (4 − 1)(
? − 2)
22 +  − 15 = (2 − 5)(
? + 3)
22 − 3 − 2 = (2 + 1)(
? − 2)
32 + 4 − 4 = (3 − 2)(
? + 2)
6 2 − 13 + 6 = 3 − 2 ? 2 − 3
15 2 − 13 − 20 = 5 + 4 ? 3 − 5
12 2 −  − 1 = 4 + 1 ?3 − 1
25 2 − 20 + 4 = 5 − ?2 2
Well Hardcore:
4 3 + 12 2 + 9 =  2 +?3
2  2 − 2 +  2 =  −?
2
2
‘Commando’ starts
to become difficult
from this question
onwards.
Simplifying Algebraic Fractions
2 2 + 4
2
?
=
2 − 4
−2
3 + 3
3
= ?
2
 + 3 + 2  + 2
2 2 − 5 − 3
2 + 1
=− ? 3
3
4
6 − 2
2
Negating a difference
− 4 −  =  −? 4
− 2 − 9 = 9 −? 2
1−
= −1 ?
−1
3 − 2 2 − 
−2
?
=
2 − 3  + 1
+1
Exercises
1
2
3
4
5
6
2 + 6  + 3
= ?
2

7
2 2 − 8
2 −2
?
=
2
 + 6 + 8
+4
4 + 8 4
= ?
3 + 6 3
8
 2 + 2 
= ?
8 + 16 8
 2 + 5 + 6  + 2
= ?
2
 +−6
−2
9
2 − 9
+3
= ?
2
2 − 7 + 3 2 − 1
2 + 10
2
= ?
2
 − 25  − 5
+3
1
= ?
2
 −9 −3
2 +  − 2  − 1
= ?
2
 −4
−2
10
11
6 2 −  − 1 3 + 1
= ?
2
4 − 1
2 + 1
2 2 + 4
9 2 − 1
× 2
=2 ?
2
3 + 7 + 2 3 − 
Algebraic Fractions
3 1
7
+
= ?
5 10 10
2 1
5
− = ?
3 4 12
How did we identify the new denominator to use?
(Note: If you’ve added/subtracted fractions before using
some ‘cross-multiplication’-esque method, unlearn it now,
because it’s pants!)
Algebraic Fractions
The same principle can be applied to algebraic fractions.
1 2
+ 2=
 

2
+ 2
2


?
+2
= 2

1
2
1
− 2
=
?
  + 2
+2
1
1
1
?
− =−
+1 
+1
1
1
2
2
?
+
−
=
3 + 6 5 + 10 15 + 30 5  + 2
5
3
4 +3
−
=
?
2 + 1 2 + 3
2 + 1 2 + 3
“To learn the secret ways of
algebra ninja, simplify
fraction you must.”
Recap
1
1
3
+
=
?1
2 + 2  + 1 2  +
1
1 1 + 
+ =
 2 
?
1 +1
1+ =
?



3
2 2 +  − 3
?
−
=
 + 1 2 + 1
 + 1 2 + 1
1
1
1+
+
=
?
2
 + +1  +1
Exercises
1
2
3
4
5
3
1
1
−
=
5 +1
2 +1
10  +?1
1
2
11
+
=
4 3 12
?
2
4
2 − 2
−
=
?
 − 1 2 − 1 2 − 1
8
1 2 + 1
2+ =
?


9
1
4 + 1
 + 2=
?

2
2
1

=
+1 +1
2
1
−1
+
=
2 − 9  + 3
 + 3 ? − 3
10
1−
2
4
2
−
=
2− 4−
2 −  ?4 − 
11
2
1
+
4 2 − 4 − 3 4 2 + 8 + 3
6
3
4
+
+1
+1
7
1
2
+5
?
−
=
 − 3 3 − 1
 − 3 3 − 1
2
3 + 7
=
?
+1 2
?
?
Completing the Square – Starter
Expand the following:
+3
2
=  2 + 6?+ 9
+5
2
? + 26
+ 1 =  2 + 10
−3
2
=  2 − 6?+ 9
+
2
? + 2
=  2 + 2
What do you notice about the coefficient of the 
term in each case?
Completing the square
Typical GCSE question:
“Express  2 + 6 in the form  + 
and  are constants.”
+3
?
2
2
+ , where 
−9
Completing the square
More examples:
2
2
?
 − 2 =  − 1 − 1
 2 − 6 + 4 =  − 3 ?2 − 5
 2 + 8 + 1 =  + 4 2? − 15
 2 + 10 − 3 =  + 5 2? − 28
2
2
 + 4 + 3 =  + 2 ? − 1
2
2
?
 − 20 + 150 =  − 10 + 50
Exercises
Express the following in the form  + 
1
2
3
4
5
6
7
8
2
+
 2 + 2 =  + 1 ?2 − 1
 2 + 12 =  + 6 ?2 − 36
 2 − 22 =  − 11 ?2 − 121
 2 + 6 + 10 =  + 3 2? + 1
 2 + 14 + 10 =  + 7 2? − 39
 2 − 2 + 16 =  − 1 2?+ 15
 2 − 40 + 20 =  − 20 ?2 − 380
2
1
1
11
2
?
 + = +
−
2
4
2
9
5
29
2
 + 5 − 1 =  + ? −
2
4
2
10
2
9
1
− 9 + 20 =  − ? −
2
4
 2 + 2 + 1 =  + 
2
?− 2 + 1
More complicated cases
Express the following in the form   + 
2
+ :
3 2 + 6 = 3  + 1?2 − 3
2 2 + 8 + 10 = 2  + 2 ?2 + 2
? 2+6
− 2 + 6 − 3 = −1  − 3
5 2 − 30 + 5 = 5  − 3 ?2 − 40
−3 2 + 12 − 6 = −3  − ?2 2 + 6
1 − 24 − 4 2 = −4  − ?3 2 + 37
Exercises
Put in the form   + 
1
2
3
4
5
6
7
2
+  or  −   + 
2 2 + 4 = 2  + 1 2?− 2
2 2 − 12 + 28 = 2  − 3 2?+ 10
3 2 + 24 − 10 = 3  + 4 2?− 58
5 2 + 20 − 19 = 5  + 2 2?− 39
− 2 + 2 + 16 = 17 −  −?1
9 + 4 −  2 = 13 −  −?2 2
2
13
3
2
1 − 3 −  =
− +
?2
4
2
2
Proofs
Show that for any integer ,
2
 +  is always even.
How many  would we need to try before
we’re convinced this is true? Is this a good
approach?
Proofs
Prove that the sum of three consecutive
integers is a multiple of 3.
We need to ensure this works for any
possible 3 consecutive numbers. What could
we represent the first number as to keep
things generic?
Proofs
Prove that odd square numbers are always 1
more than a multiple of 4.
How would you represent…
Any odd number:
2 ?+ 1
Any even number:
?
2
Two consecutive
odd numbers.
2 + 1,?2 + 3
Two consecutive
even numbers.
? +2
2 , 2
One less than a
multiple of 3.
3 ?− 1
Proofs
Prove that the difference between the squares
of two odd numbers is a multiple of 8.
Example Problems
People in the left
row work on this:
[June 2012] Prove that
2 + 3 2 − 2 − 3 2 is a
multiple of 8 for all positive
integer values of .
People in the middle
row work on this:
People in in the right
row work on this:
[Nov 2012] (In the previous
part of the question, you
were asked to factorise
2 2 + 5 + 2, which is (2 +
1)( + 2) )
[March 2013] Prove
algebraically that the
difference between the
squares of any two
consecutive integers is
equal to the sum of these
two integers.
“ is a positive whole
number. The expression
2 2 + 5 + 2 can never be a
prime number. Explain why.”
Exercises
Edexcel GCSE Mathematics Textbook
Page 469 – Exercise 28E
Odd numbered questions
Even/Odd Proofs
Some proofs don’t need algebraic manipulation. They just require us to reason
about when our number is odd and when our number is even.
Prove that 2 +  + 1 is always odd for all integers .
When  is even:
2 is  ×  = . So 2 +  + 1 is  +
 +  = .
When  is odd:
?
2 is  ×  = . So 2 +  + 1 is  +  +
 = .
Therefore  is always odd.

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