# How do you calculate the number coefficients in polynomial regression?

So I can't seem to find much on this by searching so I came here. Let's say I had 3 variables $$x_1,x_2,x_3$$ and the let's say the degree of the polynomial was $$d=2$$, I can define the length of a vector of variables as $$l$$ such that $$l=3$$ in this case. What I'm looking for is the length of a vector of coefficients that would be needed to create a polynomial regression on these variables.

Would I be right in saying that the length of the vectors of coefficients needed for polynomial regression would be $$d\times l$$?

Yes - it is $$d\times l$$. There would be 6 $$β$$ coefficients for 2*3:
$$β_1x_1 + β_2x_1^2 + β_3x_2 + β_4x_2^2 + β_5x_3 + β_6x_3^2$$