Why does binomial expansion specify raising a base to zero or one?

Question

While reviewing the Wikipedia article surrounding Pascal's triangle, I came across the following:

Pascal's triangle determines the coefficients which arise in binomial expansions. For example, consider the expansion:

$(x + y)^2 =$

$x^2 + 2xy + y^2 = $

$1x^2y^0 + 2x^1y^1 + 1x^0y^2$

However, since $x^0 = 1$, and, $x^1 = x$, why not simply write it as:

$(x + y)^2 =$

$x^2 + 2xy + y^2 = $

$x^2 + 2xy + y^2$

Is it because of the keyword "expansion", or am I missing something critical here?

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Edit: Ohhhhhhhh I get it now... Thanks to @JamesA for the answer and pointing out in the comments that I had ported it incorrectly. Had I ported it over correctly, it would've made sense and we wouldn't have this question.

Answer 1

Mathematically, there's nothing wrong with writing it the second way.

The coefficients of $1$ are written to make the coefficients match the numbers in Pascal's triangle.

The powers of zero and one make it clearer that the sum of the powers of $x$ and $y$ in each term are the same. They also make clear a quick way of writing them out, namely, increase the power of $y$ by $1$ each term and decrease the power of $x$ by $1$.

They're there more for teaching and understanding. Out in the real world you'll probably never see coefficients or powers of $1$ or $0$.