Let p : R→ R be a non-constant polynomial, that is, for every x in R, p(x) = α0 + α1x+ ... + anx, with an + 0. Prove that, if n is even then lim→∞ p(x) = +∞ and limx→-∞ p(x) = +∞ when an > 0, and lim→∞ p(x) = · 0, and limÃ→∞ p(x) = −∞ and lim¸→-∞ p(x) = when an < 0 and n is odd.

The points (1,3), (2,12) and (5,63) lie on the graph y=f (x) of some function f that is continuous everywhere. Let P2(x) denote the polynomial interpolating to f at xo = 1, x1 = 2 and x2 = 5. Assuming P2(x) was obtained via the Vandermonde method/method of undetermined coefficients, then f(3) - P2(3), which is closest to O A. -26 O B. 6 O C.-16 O D. -6 O E. 26 O F. 16

Suppose that only the points (x,y), j=1…..16, that lie on the graph y = f(x) of a function f(x) that is continuous for all x E R, are known to us. Let n be the degree of the interpolating polynomial that interpolates to f(x) at all the x₁'s for which we know the y's. Then n* =

3. а) Let xo, . . . , xn be n + 1 distinct points. Consider a function f(x) and assume there exists a polynomial p(x) of degree at most n + 1 such that p(xk) = f(xk) for k = 0,..., n, and p'(xn) = f'(xn). Show that this polynomial is unique. b) satisfies p(k) = 0 for k = 0,.., n, and p'(n) = 1. Use divided differences to find the polynomial p(x) of degree n+1 which || || .. .

Suppose f(x) has zeros at x=−3, x=4, x=5 and a y-intercept of 17.  In addition, f(x) has the following long-run behavior: as ∞x→±∞, ∞y→∞. Find the formula for the polynomial f(x) which has the minimum possible degree. Give your answer in factored form.