Subject: Math: Binomial Thrm.
First, I expanded the (x+y)-3 and got:
But I don't know how to expand the last two. I tried to and got:
(x2 + 2xy + y2)/(cubed root(x+y)) ???
I don't know if I did those right. I don't understand how this relates to the Bionomial Thrm. and what Newton found. Please help me! I'm desperate!! Thank you so much!
For n a positive integer and
the Binomial Theorem states that
Newton generalized this for n not a positive integer. The easiest example comes from looking at the identity
which is valid provided x is not 1. To see that this is valid multiply both sides by 1-x.
(Why can't x = -2 or 3 ?)
This agrees with the pattern in the statement of the binomial theorem above if a = 1, b = -x and n = -1
It was this kind of observation that led Newton to postulate the Binomial Theorem for rational exponents.
For your first example write (x+y)-3 as x-3(1+y/x)-3, expand (1+y/x)-3 using the Binomial Theorem as above:
with b = y/x and then multiply each term by x-3.
Simularly for fractional exponents. For example in your second problem write (x+y)2/3 as x2/3(1+y/x)2/3 and use the Binomial Theorem to expand (1+y/x)2/3.
You need to know some calculus to study the Binomial Theorem for rational exponents and to determine for what values of x it is true.
I hope this helps,
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