# Interpolating a function with a limited number of division operations

Let $$f(x)=-x^5$$

Approximate the value of $$f$$ in $$x=1$$, interpolating with polynomial of second degree with nodes: $$x_0=-2,\ x_1=0,\ x_2=2$$.

Then approximate it with another polynominal, interpolating with nodes: $$x_0=-2,\ x_1=0,\ x_2=2,\ x_3=-3,\ x_4=3,\ x_5=-4,\ x_6=4$$

Calculate the error in both cases.

Of course this exercise is pretty trivial, but there is an additional condition: I can use division only 4 times doing it. Additionally I am supposed to use either Lagrange's or Newton's method of interpolation.

Can you please tell me what trick can I use to reduce the number of divisions?

Computing Newton's form of the interpolating polynomial of degree $$2$$ requires $$3$$ divisions to get the necessary divided differences. You do not need any divisions to evaluate the Newton form anywhere.
In the second case, there is a unique polynomial of degree at most 6 which passes through the 7 points $$(x_i,f(x_i))$$ that are given. While $$f(x) = x^5$$ is a polynomial of degree 5 it is also true that it is a polynomial of degree at most 6 and that it passes through the 7 points. We can conclude that it is identical to the requested interpolating polynomial. Therefore there is nothing to compute and no divisions are needed.