When we expand [latex]{\left(x+y\right)}^{n}[/latex] by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand [latex]{\left(x+y\right)}^{52}[/latex], we might multiply [latex]\left(x+y\right)[/latex] by itself fifty-two times. This could take hours!
Let’s examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.
Notice that:
There are [latex]n+1[/latex] terms in the expansion of [latex]{\left(x+y\right)}^{n}[/latex].
The degree (or sum of the exponents) for each term is [latex]n[/latex].
The powers on [latex]x[/latex] begin with [latex]n[/latex] and decrease to [latex]0[/latex].
The powers on [latex]y[/latex] begin with [latex]0[/latex] and increase to [latex]n[/latex].
The coefficients are symmetric.
But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as Pascal’s Triangle.
To generate Pascal’s Triangle, we start by writing a 1. In the row below, row 2, we write two 1’s. In the 3rd row, flank the ends of the rows with 1’s, and add [latex]1+1[/latex] to find the middle number, 2. In the [latex]n\text{th}[/latex] row, flank the ends of the row with 1’s. Each element in the triangle is the sum of the two elements immediately above it.
Let’s put it side-by-side to see the connection between Pascal’s Triangle and binomial expansion. Pay attention to the coefficients!
These patterns lead us to the Binomial Theorem, which can be used to expand any binomial.
Binomial Theorem
The Binomial Theorem is a formula that can be used to expand any binomial.
Substitute [latex]n=4[/latex] into the formula. Evaluate the [latex]k=0[/latex] through [latex]k=4[/latex] terms. Notice that [latex]3x[/latex] is in the place that was occupied by [latex]x[/latex] and that [latex]-y[/latex] is in the place that was occupied by [latex]y[/latex]. So we substitute them. Simplify.
Notice the alternating signs in part b. This happens because [latex]\left(-y\right)[/latex] raised to odd powers is negative, but [latex]\left(-y\right)[/latex] raised to even powers is positive. This will occur whenever the binomial contains a subtraction sign.
