A polynomial p(x) satisfies: p(1) = p(3) = p(5) = 1 p(2) = p(4) = -1 The…

A polynomial of degree at most 4 can interpolate any five specified values at five distinct x-values, so degree 4 is possible.

Now show that degree 3 is not possible. If p(x) had degree at most 3, then q(x) = p(x) - 1 would also have degree at most 3 and would have roots at x = 1, 3, and 5. Hence q(x) must be of the form k(x - 1)(x - 3)(x - 5).

Using p(2) = -1 gives q(2) = -2. But q(2) = k(1)(-1)(-3) = 3k, so k = -2/3.

Using p(4) = -1 gives q(4) = -2. But q(4) = k(3)(1)(-1) = -3k = 2, a contradiction.

Therefore no polynomial of degree 3 or less can satisfy the conditions. Since a degree-4 interpolating polynomial exists, the minimum degree is 4.