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Binomial Expansion Reflection

Hossam Khattab, Grade 8BQatar Academy

November 3rd, 2010

Background Information

Our guiding or main questions was “is there an easy way to do 0.992?”

We discovered we could do this through binomial expansion

0.992= (1-0.01) (1-0.01)

= 12-2x1-0.01+(-0.01) (-0.01)

= 0.9801

This method is much quicker, and less hassled than using long multiplication

General Rules

For the square of the sum of two number we developed:

(a+b)2 = a2 + 2ab + b2

For the square of the difference of two numbers we developed:

(a-b)2 = a2 - 2ab + b2

Usefulness as Opposed to Traditional Multiplication

If you were an engineer 100 years ago, explain how our method may have been useful rather than just long multiplication?

Using this method is much quicker, especially for numbers that are close to tens, hundreds, etc.

The number allows you to have another way to check your answer, especially since this method is highly reliable

Long Multiplication vs. Binomial Expansion

Binomial Expansion

992= (100-1)2

= 10000 – 2x100x1 + 1

= 9801

✔

Long Multiplication

992= 899 x99 891

+8910

✗ 9801

Explanation

In the case shown previously, binomial expansion is shorter and easier, because the number is very close to a hundred

The method is useful because is takes less time, and is easier to do therefore less prone to error

It helps spread the numbers out in a way that makes multiplication very simple, because you are multiplying numbers that may have lots of digits, but the most of these digits are zeros

However…

In some situations, our method becomes cumbersome such as:

-> In situations with numbers that have many decimal places, where the amount of zeros involved becomes a problem

-> In situations where the number has many digits, and those digits are not zeros (digits including those after a decimal point)

-> In situations where it is not squaring, but it is just multiplying two large numbers

Examples

(8976)(8867)= (8000+900+70+6)(8000+800+60+7)= 80002+8000x600+8000x60+8000x7+ 900x8000+900x800+900x60+900x7+ 70x8000+70x800+70x60+70x7+6x 8000+6x800+6x60+6x7=

79590192

(96.020405)2 =(100-4+0.02+0.0004+0.000005)2= 1002+100(-4)+100x0.02+100x 0.0004+100x0.000005+100(-4)+(-4)2+(-4)x0.02+(-4)x0.0004+(-4)x0.000005 +0.02x100+0.02(-4)+0.022+0.02x 0.0004+0.02X0.000005+0.0004x100 +0.0004(-4)+0.0004x0.02+0.00042+ 0.0004x0.000005+0.000005x100+0.000005x0.02+0.000005x0.0004+ 0.0000052 =

9219.918176364025

Other Situations

Other situations where binomial expansion is not useful, and where long multiplications is definitely the way to go:

- Numbers that are not squared, but cubed, or powers larger than that

- Situations where the number has to be broken up into more than a+b (more than two numbers)

- When multiplying three or more two-digit numbers

Limitations Explained

When numbers start getting into 4, or even just 3 digits, this method becomes hard, and defeats the purpose of the mental math, because it will involve complex additions, and multiplications, and you will probably end up using long multiplication to calculate within the original calculation

When the number has to be split into more parts, the algebraic rule cannot work. Therefore you must use regular multiplication and expansion. You end up in turn, having to multiply every number by every number. This increases greatly every time you add a single digit to either of the two numbers involved.

Examples

523= (50+2) (50+2) (50+2)

= 503+50x2x2+2x50+2x50+2x2x2

= 140608

(23)(21)(15)= (20+3)(20+1)(10+5)

= 202+202+20x10+20x1+20x5+3x20+3x1+3x10+3x5

=7245

2332=(200+30+3) (200+30+3)

= 2002+200x30+200x3+30x200 +302+30x3+3x200+3x30+32

= 54289

Conclusion

Binomial Expansion is very useful in general for multiplying 2 or 3 digit numbers, and squaring them, which would usually be difficult or would require long multiplication

It is a shortcut method, to reduce working for products where long multiplication would otherwise be necessary, however it cannot completely replace long multiplication, simply because it starts to get confusing with many decimals, large numbers, and larger powers.