If x² is a factor of the polynomial f(x), then it must also be true that f(0) = 0 and f'(0) = 0.
This is because if x² divides f(x), then the polynomial can be expressed in the form:
f(x) = x²·g(x),
where g(x) is another polynomial. Since x² is a multiple of x, it implies that f(0) = 0, as substituting x = 0 into the equation gives f(0) = 0²·g(0) = 0.
Additionally, the presence of x² as a factor means that the polynomial touches the x-axis at x = 0, creating a double root. Consequently, the first derivative f'(x) must also equal zero at x = 0:
f'(x) = 2x·g(x) + x²·g'(x),
which leads to f'(0) = 0. Therefore, when x² is a factor of f(x), both f(0) = 0 and f'(0) = 0 must hold true.