Binomial Expansions – Reflection Criterion D. By: Mohammed Shooshtarian. Introduction.
Binomial Expansions – Reflection Criterion D
An Image/Link below is provided (as is) to download presentation
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
In this power point I will be explaining in detail and giving many examples. 1) How our method may have been useful to an engineer 100 years ago rather than just using long multiplication, 2) When the method wouldn’t come in handy and its best to use long multiplication and 3) I will give detailed explanations and examples of where long multiplication is more efficient than our expansion method.
Engineers didn’t have the benefits that we had with technology until the late 1960’s. Since engineers didn’t have calculators at the time, 100 years ago, it was easier and faster to multiply numbers with only 1 unit e.g. 10, or 0.01. For example: (12)2 is much easier and faster to work out when you break it down to (10+2) (10+2), using the FOIL method or the faster method which is shown on the next page. This is because 100 years ago it was important for an engineer to have a method that works using a formula, because as I said, they didn’t have calculators so they would need to use special procedures in order to calculate faster rather than using long multiplication which would then slow them down.
General Rule: (a+b)2 = a2 + 2ab + b2 and (a-b)2 = a2 - 2ab - b2
(10 + 2) (10 + 2)
102 + 2 x 10 x 2 + 22
100 + 40 + 4
This can be solved in your head easily, instead of using long multiplication which would take much longer and would be more complicated.
(90 + 5)2 =
902 + 2 x 90 x 5 + 52 =
8100 + 900 + 25 =
If the second digit is 5 or less and it’s a 2 digit number then you should use the general rule by squaring the sum
(100 – 3) =
1002 – 2 x 100 x 3 – 32 =
100000 – 600 – 6 =
If the second digit is 6 or more and it’s a 2 digit number then you should use the general rule by squaring the difference
If you have too many digits its best to multiply by long hand.
For example (1234)2 is hard to work out using binomial expansions because (1230 + 4) is a really difficult number, and it is best to use long multiplication when it’s a long number with no zeros.
E.g. 1: (10.02)2 =(10 + 0.02)2 = 100 + 2 x 10 x 0.02 + (0.02)2 =100 + 0.40 + 0.0004 = 100 + 0.40 + 0.0004 = 100.4004, it doesn’t matter how many decimal places there are but it matters if there are zeros included in the number, instead of the zeros if the numbers were 1’s or 2’s or 3’s it would be much harder to solve, and it would be more efficient to use long multiplication.
Numbers that have 4 or maybe even 3 digits are best to work out by using long multiplication. This depends if the number includes a zero, if it does then I wouldn’t consider using long multiplication, but if it didn’t it would be best to use long multiplication.
E.g. 2: (783) 2 = (780 + 3) 2 = 7802 + 2 x 780 x 3 + 32
As you can see the method that I am using is much more complicated than just simply multiplying 783 by 783 with long multiplication.
Long multiplication would be a better way of calculating if the number you are squaring has 3, 4, 5, 6, 7, 8 etc, and more digits and includes a couple of zeros, it doesn’t matter if the zero(s) are in between the numbers or if they at the end, it will be much easier to binomially expand.
These examples are also easy to do using long multiplication, although you might fall off track on the zeros and it might mess up the whole answer.
If the number of non-zero digits is more than 2 or if the number of non-zero digits is 3 and both of the digits are bigger than 5 then it would be best to use long multiplication.
E.g. 3: (178)2 = (170 + 8)= 1002 + 2 x 170 x 8 + 82 is hard to do in your head, people 100 years ago didn’t have any calculators to multiply 170 by 2 or even the total of 170 x 2 by 8, so they would use long multiplication, instead of doing all of this you could just do 178 multiplied by 178 with long multiplication.
E.g. 4: 178
x 178 = 31684
If the number didn’t have a zero in it and it were a 3 or 4 digit number it would be really hard to work out without a calculator
E.g. 5: (1423)2 =(1420 + 3) = 14202 + 2 x 1420 x 3 + 32 = 2 x 1420 (which might be easy to do in your head) = 2840 x 3 (which is hard to calculate in your mind, so your better of multiplying 1423 by 1423 using long multiplication)
Pascal Triangle Theory
The Pascal theory is another much helpful way of calculating binomial expansion.
As you can see in the picture the third line is 1 2 1 which represents (a + b)2 and the fourth line represents (a + b)3 and the fifth line represents (a + b)4 and so on.
E.g. 1: (121) (a + b)2 = (1)a2 + 2ab + (1)b2
E.g. 2: ( 1331) (a + b)3 =(1)a3+ 3a2b + 3ab2+ (1)b3
E.g. 3: (14641) (a + b)4 = (1)a4 + 4a3b + 6a2b2 + 4ab3 + (1)b4
Looking at the general rule and the long multiplication method when working out exponentials, we can see that sometimes it is better to use one method rather than the other. The bigger the number the better it is to use long multiplication, the bigger the number and the more zeros included in the number it is better to use binomial expansions.