Question

Explain why all solutions must be non-zero for this polynomial function. An algebraic argument must be provided for full credit. Graphing by hand or with technology will be granted partial credit unless the evidence on the graph is supported with algebra.

Answer 1

For a polynomial function, if any of its solutions were zero, that would mean that its corresponding factor is zero. Thus, the polynomial would have a factor of zero, which would make the entire polynomial zero. In other words, if there were a solution of zero, that solution would be a root of the polynomial function.

However, a polynomial function can only have a finite number of roots, which is equal to its degree. Therefore, if a polynomial function had a solution of zero, that would mean that it has one more root than its degree, which is impossible. This is because a polynomial function with degree n can have at most n roots.

Therefore, all solutions of a polynomial function must be non-zero, or in other words, the function must have no roots of zero. This ensures that the polynomial function has the correct number of roots, which is equal to its degree.

This argument applies to all polynomial functions, regardless of their degree or coefficients. Therefore, it is a general property of polynomial functions that all of their solutions must be non-zero.