you won't really need the binomial theorem for methods, but anyway:
1. the binomial theorem
a binomial is a polynomial with 2 terms, you can use the binomial theorem for expressions of the form
^n)
.
the binomial theorem is used to expand when n is not a small number.
2. formulas
(n-2)...(2)(1))
!} )
^n &= \begin{pmatrix} n \\ n \end{pmatrix}a^nb^0 + \begin{pmatrix} n \\ n-1 \end{pmatrix} a^{n-1}b^1 + ... + \begin{pmatrix} n \\ 1 \end{pmatrix} a^1b^{n-1} + \begin{pmatrix} n \\ 0 \end{pmatrix} a^0b^n <br />\\ &= \begin{pmatrix} n \\ n \end{pmatrix}a^n + \begin{pmatrix} n \\ n-1 \end{pmatrix} a^{n-1}b^1 + ... + \begin{pmatrix} n \\ 1 \end{pmatrix} a^1b^{n-1} + \begin{pmatrix} n \\ 0 \end{pmatrix} b^n\end{aligned})
3. using the binomial theorem
by expanding:
^2 &= (a+b)(a+b) <br />\\ &= a^2 + ab + ab + b^2 <br />\\ &= a^2 + 2ab + b^2<br />\\ n=3: <br />\\ (a+b)^3 &= (a+b)(a+b)(a+b) <br />\\ &= (a^2+ 2ab + b^2)(a+b) <br />\\ &= a^3 + 3a^2b + 3ab^2 + b^3 ... \end{aligned})
by using the binomial theorem:
^2 &= \begin{pmatrix} 2 \\ 2 \end{pmatrix}a^2b^0 + \begin{pmatrix} 2 \\ 1\end{pmatrix}a^1b^1 + \begin{pmatrix} 2\\ 0 \end{pmatrix}a^0b^2<br />\\ &= a^2 + 2ab + b^2 \end{aligned})
try that with n=3
4. other uses of the binomial theorem
suppose you wanted to find the coefficient of the term a^7 in (a+2)^8 - you would do:
thus, the coefficient of the
term is 16.
you're also not restricted to using it for polynomials (to some extent)
if you wanted to find the coefficient of the
term in
^3)
first, you would try to form a binomial:
^3 = \frac{(a^3+1)^3}{a^3})
now, all you have to do is 'locate' the
term (find which values of n and r to apply the binomial theorem)
in this case, the
in the bottom needs to cancel an
on the top to give
, thus, n=3 and r = 2 (since
)
so,
!} = \frac{6}{2} = 3)
5. other information
it doesn't matter whether you choose to do
(going downwards) or
(going upwards) for the second term and onwards because they are the same value.
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