Binomial Expansions

 

Binomial Expansions

By algebraic definition, binomials are the sum of two objects. Let $(x+y)$ be any binomial (that is to say, whatever the two objects being added together are, let $x$ be the first and $y$ be the second).Binomial expansion refers to the result you get when you raise a binomial to a power. As much as we may wish it so,$$(x+y)^n \ne x^n + y^n$$This can be instantly proven for $n=2$:$$(x+y)^2 = x^2 + 2xy + y^2 \ne x^2 + y^2$$The hard truth is that, in order to multiply binomials, you need to distribute (FOIL, in the case of $n=2$). That is, for example,$$(x+y)^2 = (x+y)(x+y)$$This quickly becomes tedious and annoying - even multiplying three binomials together seems like a drag.$$(x+y)^3 = (x+y)(x+y)(x+y)$$However, Pascal's Triangle gives a magnificent shortcut for this process! If you want to multiply $(x+y)^n$, follow these guidelines:
  • The variable part of the first term is $x^n$
  • The variable part of the next term is $x^{n-1} y$
  • Each sequential term's variable piece decreases the exponent on $x$ and increases the exponent on $y$. i.e. the third term is of the form $x^{n-2} y^2$
  • Keep going until you have decreased $x$ to nothing, at which point your last term will be $y^n$
  • If you have $(x+y)$ then all terms are positive, and if you have $(x-y)$ then the first term is positive, the second term is negative, and each term after that alternates
  • Finally, and significantly - the coefficients of each term will come from the $n+1$ row in Pascal's Triangle
Recall that Pascal's Triangle » looks like this:Here are three examples. See if you can line up the guidelines above with the results. When you know the pattern, you don't actually have to do the tedious multiplication to get the answer!$$(x+y)^5$$
 
$$(x-y)^7$$
 
$$(2a + 3b)^4$$