Is there a way to reduce this polynomial?

Question

The following cubic polynomial $P(x)$ goes through the points $(x_0, y_0), (x_1, y_1), (x_2, y_2), (x_3, y_3)$:

$P(x)=\frac{y_0(x-x_1)(x-x_2)(x-x_3)}{(x_0-x_1)(x_0-x_2)(x_0-x_3)}+\frac{y_1(x-x_0)(x-x_2)(x-x_3)}{(x_1-x_0)(x_1-x_2)(x_1-x_3)}+\frac{y_2(x-x_0)(x-x_1)(x-x_3)}{(x_2-x_0)(x_2-x_1)(x_2-x_3)}+\frac{y_3(x-x_0)(x-x_1)(x-x_2)}{(x_3-x_0)(x_3-x_1)(x_3-x_2)}$

Is there a reasonable way to put this polynomial into the form

$$P(x)=a_3x^3+a_2x^2+a_1x+a_0$$

so the coefficients $a_i$ are in terms of the parameters $x_i$ and $y_i$? I don't have many CAS softwares, the one from geogebra gives me an incredibly ugly result and I can't even put the whole expression into the free version of wolfram, so any suggestions?

Note: one of my classmates says he simplified that expression "by hand", is that possible?

Answer 1

Is there a reasonable way

That depends on how reasonable you consider "reasonable". For example, you could combine all the fractions so that they're over the common denominator $(x_1-x_0)(x_2-x_0)(x_3-x_0)(x_2-x_1)(x_3-x_1)(x_3-x_2)$ and then multiply out whatever you get on top. There might be a better way, but I don't know of it.

the one from geogebra gives me an incredibly ugly result

The result probably is ugly. (Have you tried doing this for e.g. a quadratic or even a linear polynomial? I'd be willing to bet that's probably ugly too.)

one of my classmates says he simplified that expression "by hand", is that possible?

Sure, it's possible. It's not even conceptually difficult - it's just really long and tedious.