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 SquaresWhat is the difference of two squares?

The law states that if an expression is multiplied by its conjugate, we get the difference of and .

for .

This rule has many uses to help make calculations easier.

The Difference of Two Squares for .

Let’s use this identity to help us make some numerical calculations.Example

Calcuate .

Consider the above for .(50+1 ) (50−1 )

¿502−12

¿2500−1¿2499

The Difference of Two Squares

for .

We also need to consider the opposite principle.

Example

Calcuate .

Consider the above for .(45+5 ) (45−5 )

¿50×4 0¿2000

Answers6399 396 2491

2451 8019 3536

999900 8096 48.91

7𝑎𝑛𝑑 9

600 1800 6600

900000 3800000 −9000

Numerical ProofsBefore we look at proofs, we should recall some vocabulary.

Odd - A number in the form where . (1, 3, 5, 7, …)

Even - A number in the form where . (2, 4, 6, 8, …)

Consecutive Integers - A pair of integers where .

(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 where .)

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 ProofsLet’s try a simple proof and divide the steps up.

ExampleProve that an odd number squared is odd.Step 1

Let an odd number where .Step 2

If

¿2(2𝑛2−2𝑛)+1Step 3

As is even, must be odd. If is odd, must be odd.

Numerical ProofsLet’s try another.ExampleIf two integers are odd, their sum is even.Step 1

Consider two integers where and .

Step 2

The sum of and , Step 3

As is a factor of , is even.

If and are odd, must be even.

¿2𝑎+2𝑏−2=2(𝑎+𝑏−1)

Answers – Easy (Top)Let an odd number be in the form where . where . As can be written in the form , , 7 is odd.

Let an even number be in the form where . where . As can be written in the form , , 12 is even.

Let an even number where .If , As is a factor of , is even. An even number squared is even.

Answers – Easy (Bottom)Let an even number where . If , As is even, is half of an even number. , Half of an even number is an integer.

Consider two integers where and . The sum of and , As is a factor of , is even. If two integers are even, their sum is even.

Let be an odd number where . As , As is odd, where is odd. The positive root of a square odd number is odd.

Answers – Hard (Top)Let a pair of consecutive odd numbers , , . . As is a factor of , the product of two consecutive odd numbers plus 1 is a multiple of 4.Consider two integers where and . The sum of and , As is a factor of , is odd. The sum of an odd and even number is odd.

Consider two integers where and . The product of and , As is a factor of the expression is even. is odd. The product of two odd numbers is odd.

Answers – Hard (Bottom)

Consider two integers where and . The product of and , As is a factor of, the expression is even. The product of an odd and even number is even.

Let an odd number be in the form where .By substitution, As is a factor of, the expression is even. If is odd, is even.