List all degree 42 polynomials that pass through the eleven points with a constant of 42

Question

I am having trouble figuring out (1) how to conceptualize this problem and (2) how to approach it.

The question is regarding interpolation: List all degree 42 polynomials that pass through the eleven points $(-5,5),(-4,4),...,(4,-4),(5,-5)$ AND have a constant of 42.

I'm guessing that the approach to this problem may involve generalizing the Lagrange interpolation method, but I am not sure how to do this nor account for the constant of 42.

How can I list all of the degree 42 polynomials listed here that have a constant of 42?

Answer 1

Use the unconventional notation for the desired polynomial, $t$ for the variable, and $x_i$ for the coefficient of $t^i$, that is: $$f(t) =x_0 + x_1t+x_2t^2+\cdots x_{42}t^{42}\qquad(*)$$.

You want this polynomial to pass through some 11 points, say $(a_1,b_1),\ldots(a_{11},b_{11})$. This means $f(a_i)=b_i$. Substituting $t=a_i$ in $(*)$ we get 11 linear equations in the 42 variables $x_i$.

IN general this will have infinitely many solutions (when consistent). ANy solutions for the $x_i$'s give rise to one such polynomial you desire.