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Binomial Expansions. By: Alia Al-Khater. What are Binomial Expansions?. Binomial expansions are easier ways of performing complex numerical calculations easily. The general rule for expanding binomials is show below: (a+b) 2 = a 2 + 2ab + b 2 and (a-b) 2 = a 2 – 2ab + b 2.
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Binomial Expansions By: Alia Al-Khater
What are Binomial Expansions? • Binomial expansions are easier ways of performing complex numerical calculations easily. • The general rule for expanding binomials is show below: (a+b)2 = a2 + 2ab + b2and(a-b)2 = a2 – 2ab + b2
Binomial Expansions For example a binomial expansion is: (x + 5)5 = (x + 5)(x + 5)(x + 5)(x + 5)(x + 5) (x + 5)(x + 5)(x + 5)(x + 5)(x + 5) = (x + 5)(x + 5)(x + 5)(x2 + 10x + 25) = (x + 5)(x + 5)(x3 + 15x2 + 75x + 125) = (x + 5)(x4 + 20x3 + 150x2 + 500x + 625)
100 Years Ago • 100 years ago when there was no calculators, the general rule would be much easier than using long multiplication. Simple binomial expansions could’ve been done in somebody’s head and it would have taken much less time than having to do it with long multiplication.
Simple & Complex Expansions • A simple binomial expansion would be easy and quick to do in someone’s head, but if you were to do a more complex binomial expansion it would be more accurate to use long multiplication than the rule we have came up with. Simple : (9)2 = (10-1)(10-1) Complex: (0.487)2 = (1-0.513)(1-0.513) = 100 – 1 – 10 + 1 = 1 – 0.513 – 0.513 + 0.263169 = 90 = 0.237169
Complex Numerical Calculations • At the point where the number reaches beyond three decimals points, it is better to use long multiplication. It would be easier to do it that way, rather than with the general rule that we have came up with. Also large numbers with over 3 digits can be hard when using this method, especially if you want it done in your head. You still can always use this method, even though it may take a bit longer than with a simple binomial expansion.
Long Multiplication Expansion Method • There are times where long multiplication is better to use to be more accurate. For example, 12345 = (1200+34)(1200+34) is not a simple problem to expand. Also, with that you would have to multiply your answer by 5. I think doing long multiplication would be easier to use and much more accurate.
Long Multiplication Expansion Method • 8943 = (900-6)(900-6). For this example I also think it would be better to use long multiplication because its cubed. If you used long multiplication your answer would be: • 894 x 894=799,236 x 894 = 714,516,984. These are large numbers and using long multiplication would be easier for this equation, since the problem is much more complex than a simple binomial expansion.
