Recently, I was watching Quanta Magazine's The Biggest Project in Modern Mathematics YouTube video. I was thrown off by the expansion of $(1-x)^{24}$ to $(1-24x+276x^2)$ at around timestamp 3:40. Is this an error? Or are they using some non-standard notation? Or completely glossing over a lot of details?
Of course, WolframAlpha confirms that the expansion of $(1-x)^{24}$ is not $(1-24x+276x^2)$, but something more unwieldy. $(1-x)^{24}$ and $(1-24x+276x^2)$ don't even have the same roots!
The video seems like it's targeted to a general audience, so if they are using some super non-standard notation, I would have hoped they had pointed it out.
I asked in the video comments, but no one answered, so I'm asking here!
You're right (of course) that $(1-x)^{24}$ doesn't equal $1-24x+276x^2$. However, the expansion that's happening here is a Taylor series: taking the function $$x(1-x)^{24}(1-x^2)^{24}(1-x^3)^{24}\cdots$$ and expanding it as $x+a_2x^2+a_3x^3+\cdots$, where $a_2,a_3,\dots$ are coefficients (i.e. expanding around $0$ -- this information is useful for approximating the function for small $x$). This means that, to compute any particular $a_i$, you don't necessarily need the whole expansion of $(1-x)^{24}$, just the first few terms. This is presumably why it's expanded as $1-24x+276x^2$ in the video, although it would be clearer if it were written $1-24x+276x^2-\cdots$, to make it clear that there are more terms, and they just didn't want to take the space to write them down.