# How can you use the binomial theorem to expand #(y-2)^7# ?

##### 1 Answer

#### Explanation:

By the binomial theorem:

#(a+b)^n = ((n),(0))a^n + ((n),(1))a^(n-1)b + ((n),(2))a^(n-2)b^2 + ... + ((n),(n))b^n#

where

Rather than mess with all those factorials directly, we can pick out the appropriate row of Pascal's triangle, which in our example is the row beginning

So we find:

#(a+b)^7 = a^7+7a^6b+21a^5b^2+35a^4b^3+35a^3b^4+21a^2b^5+7ab^6+b^7#

In our example,

The easiest way is probably to write out some sequences to construct our coefficients...

Here's the row from Pascal's triangle:

#1, 7, 21, 35, 35, 21, 7, 1#

Here are powers of

#1, 2, 4, 8, 16, 32, 64, 128#

Multiply the two sequences together to get:

#1, 14, 84, 280, 560, 672, 448, 128#

Then alternate the signs:

#1, -14, 84, -280, 560, -672, 448, -128#

These are the coefficients we need. So we have:

#(y-2)^7 = y^7-14y^6+84y^5-280y^4+560y^3-672y^2+448y-128#