S

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

-This means the first term squared, plus the product of the first and second term times two, plus the last term squared

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

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

-This is slightly different, where the first term is still squared, however the product of the first and second term squared is then subtracted from the first term squared, then the last term squared is added

Advantages 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?

An engineer 100 years ago could rely on this method as a shortcut. Since an engineer 100 years ago would not have had a calculator they would be using pencil and paper.

Shortcuts like this would therefore be important, as they increase efficiency and time productivity

This is especially important to someone like an engineer who is bound by a time schedule or a limit

To an Engineer….

For someone like an engineer, this method is: Useful, because it is very straightforward, and has

many applications Reliable, because it has little steps, with less room

for error. For an engineer reliability is important Because it has a lower chance of error, it is accurate,

and in engineering like building houses, it needs to be, because people are depending on this

Long Multiplication vs. Binomial Expansion (992)

Binomial Expansion

992= (100-1)2

= 10000 – 2x100x1 + 1

= 9801

✔

Long Multiplication

992= 899 x99 891

+8910

✗ 9801

Explanation

This method lets us write the number as either the sum or difference of two numbers

The method allows you to multiply numbers you are comfortable with, in this case, 100 and 1.

Alternative to long multiplication, because the method is different, so it is shorter.

This is because writing the number as a sum or difference you end up with the same result, but during the process the multiplication is different

Further Explained…

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

If you are not using the algebraic method, you are still able to use this method, by following the distributive law

However…

In some situations, our method becomes cumbersome and messy such as: When the numbers have many decimal places. This

added to the whole numbers already to the left can get confusing

At the end, it is harder to know where to place the decimal point

Also, there are decimals which may have zeros, which are normally easily to multiply, but will later on give you more complicated decimals to add and multiply

Examples

96.022= (90+6+0.02)2

= 902+90x6+90x0.02+6x90+ 62+6x0.02+0.02x90+0.02 x6+0.022

= 8100+540+1.8+540+36 +0.12+1.8+0.12+0.04

= 9219.8404

96.02x96.0219204

546120086418000

9219.8404

Further Examples

83.42= (80+3+0.4)2

= 802+80x3+80x0.4+3x80+32 +3x0.4+0.4x80+0.4x3+ 0.42

= 6400+240+32+240+9+0.12+32+0.12+0.16

= 6955.56

83.4x83.43336

2502066720

0

6955.56

Other Situations

Other situations where using long multiplication is probably a better option than binomial expansion Number to powers greater than two, i.e. cubed, to

the power four, etc. When numbers must be broken up into three ore

more parts, such as three digit numbers When multiplying three or more two-digit numbers Numbers that have 3 or more digits

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

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

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

= 400+400+200+20+100+60+3+30+15

=7245

23x2123

460483x15

24154830

7245

Further Examples Pt. 2

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

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

= 125000+200+100+100+8

= 140608

52x52104

26002704 x525408

135200

140608

P a s c a l ’ sTRIANGLE

-Is formed by adding adjacent numbers and writing the answer in the line under in a brick-fashion-First diagonal is formed from ones. Second is formed from counting numbers and the third is formed from triangular number. The white area is tetrahedral numbers- Horizontal Sums are equal to two to the power of the row. This is a very useful as an alternate method to finding large powers, instead of both long multiplication and binomial expansion

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.