why all solutions must be non-zero for this polynomial ? f(x)=10x^5-15x^4+12x^3-18x^2+2x-3

Question

why all solutions must be non-zero for this polynomial ? f(x)=10x^5-15x^4+12x^3-18x^2+2x-3

Answer 1

Alternatively, you can use a calculator and graph it. All solutions must have y = 0, when the graph crosses the x-axis. There is only one real solution occurring at ( 3/2, 0). So the only solution is non-zero.

Answer 2

Assuming that you mean "solutions" are real roots, then since f(x) has a nonzero constant term, then x = 0 can't be a root since f(0) = -3. Thus, if x is a real root, then x must be nonzero.