Download Applications of Expansion and Factorisation

Applications of
Expansion and
Factorisation
SLIDESHOW 17
MATHEMATICS
MR SASAKI
ROOM 307
Objectives
•Use the difference of two squares to make
certain numerical calculations
•Be able to prove certain simple numerical
facts
The Difference of Two
Squares
What is the difference of two squares?
The law states that if an expression 𝑎 + 𝑏 is
multiplied by its conjugate, we get the
difference of 𝑎2 and 𝑏 2 .
𝑎 + 𝑏 𝑎 − 𝑏 ≡ 𝑎2 − 𝑏 2 for ∀ 𝑎, 𝑏 ∈ ℝ.
This rule has many uses to help make
calculations easier.
The Difference of Two
Squares
𝑎 + 𝑏 𝑎 − 𝑏 ≡ 𝑎2 − 𝑏 2 for ∀ 𝑎, 𝑏 ∈ ℝ.
Let’s use this identity to help us make some
numerical calculations.
Example
Calcuate 51 × 49.
Consider the above for 𝑎 = 50, 𝑏 = 1.
51 × 49 = 50 + 1 50 − 1
= 502 − 12
= 2500 − 1 = 2499
The Difference of Two
Squares
We also need to consider the opposite
principle.
𝑎2 − 𝑏 2 ≡ 𝑎 + 𝑏 𝑎 − 𝑏 for ∀ 𝑎, 𝑏 ∈ ℝ.
Example
Calcuate 452 − 52 .
Consider the above for 𝑎 = 45, 𝑏 = 5.
452 − 52 = 45 + 5 45 − 5
= 50 × 40
= 2000
Answers
6399
396
2491
2451
8019
3536
999900
8096
48.91
7 𝑎𝑛𝑑 9
600
1800
6600
900000
3800000
−9000
Numerical Proofs
Before we look at proofs, we should recall
some vocabulary.
Odd - A number in the form 2𝑛 − 1 where
𝑛 ∈ ℤ+ . (1, 3, 5, 7, …)
Even - A number in the form 2𝑛 where 𝑛 ∈ ℤ+ .
(2, 4, 6, 8, …)
Consecutive Integers A pair of integers 𝑛, 𝑛 + 1 where 𝑛 ∈ ℤ.
(for example 15 and 16.)
Proofs (Format)
When you prove something, there should be
three steps:
Step 1 - A statement about what you have to prove.
(eg: Let an even number be a number be in
the form 2𝑛 where 𝑛 ∈ ℤ+ .)
Step 2 - Perform the necessary calculation. Write
what you need to in the correct form.
Step 3 - State that as it’s now in the correct form,
the proof for the original expression is
now complete.
Numerical Proofs
Let’s try a simple proof and divide the steps up.
Example
Prove that an odd number squared is odd.
Step 1
Let an odd number 𝑎 = 2𝑛 − 1 where 𝑛 ∈ ℤ+ .
Step 2
If 𝑎 = 2𝑛 − 1, 𝑎2 = 2𝑛 − 1 2 = 4𝑛2 − 4𝑛 + 1
2
=
2(2𝑛
− 2𝑛) + 1
Step 3
As 𝑎2 = 2 2𝑛2 − 2𝑛 + 1 is in the form 2𝐴 + 1
where 𝐴 ∈ ℤ+ , 𝑎2 must be odd. ∴ If 𝑎 is odd, 𝑎2
must be odd.
Numerical Proofs
Let’s try another.
Example
If two integers are odd, their sum is even.
Step 1
Consider two integers 𝑥, 𝑦 where 𝑥 = 2𝑎 − 1, 𝑦 =
2𝑏 − 1 and 𝑎, 𝑏 ∈ ℤ+ .
Step 2
The sum of 𝑥 and 𝑦, 𝑥 + 𝑦 = 2𝑎 − 1 + (2𝑏 − 1)
= 2𝑎 + 2𝑏 − 2 = 2(𝑎 + 𝑏 − 1)
Step 3
As 𝑥 + 𝑦 is in the form 2𝐴 where 𝐴 ∈ ℤ+ , 𝑥 + 𝑦 is
even.
∴ If 𝑥 and 𝑦 are odd, 𝑥 + 𝑦 must be even.
Answers – Easy (Top)
Let an odd number be in the form 2𝑛 − 1 where
𝑛 ∈ ℤ+ . 7 = 2 ∙ 4 − 1 where 4 ∈ ℤ+ . As 7 can be
written in the form 2𝑛 − 1 where 𝑛 ∈ ℤ+ , 7 is odd.
Let an even number be in the form 2𝑛 where
𝑛 ∈ ℤ+ . 12 = 2 ∙ 6 where 6 ∈ ℤ+ . As 12 can be
written in the form 2𝑛 where 𝑛 ∈ ℤ+ , 12 is even.
Let an even number 𝑎 = 2𝑛 where 𝑛 ∈ ℤ+ .
If 𝑎 = 2𝑛, 𝑎2 = 2𝑛 2 = 4𝑛2 = 2 2𝑛2 .
As 𝑎2 is in the form 2𝐴 where 𝐴 ∈ ℤ+ , 𝑎2 is
even. ∴ An even number squared is even.
Answers – Easy (Bottom)
Let an even number 𝑎 = 2𝑛 where 𝑛 ∈ ℤ+ . If 𝑎 = 2𝑛,
𝑎
𝑎
= 𝑛 ⇒ ∈ ℤ+ as 𝑛 ∈ ℤ+ .∴ Half of an even number
2
2
is an integer.
Consider two integers 𝑥, 𝑦 where 𝑥 = 2𝑎, 𝑦 = 2𝑏 and
𝑎, 𝑏 ∈ ℤ+ . The sum of 𝑥 and 𝑦, 𝑥 + 𝑦 = 2𝑎 + 2𝑏 =
2 𝑎 + 𝑏 . As 𝑥 + 𝑦 is in the form 2𝐴 where 𝐴 ∈ ℤ+ , 𝑥 +
𝑦 is even.∴ If two integers are even, their sum is even.
Let 𝑎 be an odd number where 𝑎 = 2𝑛 − 1, 𝑛 ∈ ℤ+ . As
𝑎2 = 2𝑛 − 1 2, 𝑎2 = 4𝑛2 − 4𝑛 + 1 = 2 2𝑛2 − 2𝑛 + 1.
As 𝑎2 is in the form 2𝐴 + 1 where 𝐴 ∈ ℤ+ , 𝑎2 is odd. ∴
The positive root of a square odd number is odd.
Answers – Hard (Top)
Let a pair of consecutive odd numbers 𝑎 = 2𝑛 − 1, 𝑏 =
2𝑛 + 1, 𝑛 ∈ ℤ+. 𝑎𝑏 + 1 = 2𝑛 − 1 2𝑛 + 1 + 1 = 4𝑛2.
As 𝑎𝑏 + 1 is in the form 4𝐴 where 𝐴 ∈ ℤ+, the product of
two consecutive odd numbers plus 1 is a multiple of 4.
Consider two integers 𝑥, 𝑦 where 𝑥 = 2𝑎 − 1, 𝑦 = 2𝑏 and
𝑎, 𝑏 ∈ ℤ+. The sum of 𝑥 and 𝑦, 𝑥 + 𝑦 = 2𝑎 − 1 + 2𝑏 =
2 𝑎 + 𝑏 − 1. As 𝑥 + 𝑦 is in the form 2𝐴 − 1where 𝐴 ∈ ℤ+,
𝑥 + 𝑦 is odd. ∴ The sum of an odd and even number is odd.
Consider two integers 𝑥, 𝑦 where 𝑥 = 2𝑎 − 1, 𝑦 = 2𝑏 − 1
and 𝑎, 𝑏 ∈ ℤ+. The product of 𝑥 and 𝑦, 𝑥𝑦 = (2𝑎 −
Answers – Hard (Bottom)
Consider two integers 𝑥, 𝑦 where 𝑥 = 2𝑎 − 1, 𝑦 =
2𝑏 and 𝑎, 𝑏 ∈ ℤ+ . The product of 𝑥 and 𝑦, 𝑥𝑦 =
2𝑏 2𝑎 − 1 = 2(2𝑎𝑏 − 𝑏). As 𝑥𝑦 is in the form 2𝐴
where 𝐴 ∈ ℤ+ , the expression is even. ∴ The product
of an odd and even number is even.
Let an odd number 𝑛 be in the form 𝑛 = 2𝑎 − 1
where 𝑎 ∈ ℤ+ .
By substitution, 3𝑛 + 7 = 3 2𝑎 − 1 + 7 = 6𝑎 +
4 = 2 3𝑎 + 2 . As 3𝑛 + 7 is in the form 2𝐴 where
𝐴 ∈ ℤ+ , the expression is even.
∴ If 𝑛 is odd, 3𝑛 + 7 is even.