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Binomial Expansion

Binomial Expansion refers to expanding an expression that involves two terms added together and raised to a power, i.e. (x+y)^n. In the simple case where n is a relatively small integer value, the expression can be expanded one bracket at a time:

Example 1
(x+y)^3=(x+y)(x+y)(x+y)=(x^2+2xy+y^2)(x+y)=x^3+3x^2y+3xy^2+y^3
Example 2
(x+y)^4=(x+y)^3(x+y)=x^4+4x^3y+6x^2y^2+4xy^3+y^4 using the expansion above.
Expanding (x+y)^n by hand for larger n becomes a tedious task. The Edexcel Formula Booklet provides the following formula for binomial expansion:

(a+b)^n=a^n+\left(\begin{array}{c}n\\1\end{array}\right)a^{n-1}b+\left(\begin{array}{c}n\\2\end{array}\right)a^{n-2}b^2+...+\left(\begin{array}{c}n\\r\end{array}\right)a^{n-r}b^r+...b^n

where

\left(\begin{array}{c}n\\r\end{array}\right)=\frac{n!}{(n-r)!r!}

for when n\in{\mathbb N}, i.e for when n is a positive integer.

Example 3 – Find the first three terms, in descending powers of x, in the binomial expansion of (2x+4)^5.
This can be done using the formula above. Perform a direct substitution as follows: a=2x, b=4 and n=5 and take the first three terms.

(2x+4)^5=(2x)^5+\left(\begin{array}{c}5\\1\end{array}\right)(2x)^{5-1}(4)+\left(\begin{array}{c}5\\2\end{array}\right)(2x)^{5-2}(4)^2+...
Note that
\left(\begin{array}{c}5\\1\end{array}\right)=\frac{5!}{(5-1)!1!}=\frac{120}{20\times 1}=5
and
\left(\begin{array}{c}5\\2\end{array}\right)=\frac{5!}{(5-2)!2!}=\frac{120}{6\times 2}=10
and so the formula becomes
(2x+4)^5=(2x)^5+5\times(2x)^{4}(4)+10\times(2x)^{3}(4)^2+...=32x^5+320x^4+1280x^3+...

Note that if the question says ascending powers of x, then a and b can be switched over so that the powers of x are increasing instead.