Ind the zeros and their multiplicities. Consider using Descartes' rule of signs and the upper and lower bound theorem to limit your search for rational zerd \( f(x)=14 x^{3}-55 x^{2}+42 x-9 \) Part: \( 0 / 2 \) Part 1 of 2 Select "None" if applicable. The zeros of \( f(x) \) : \( \square \)

To find the zeros of \( f(x) = 14x^3 - 55x^2 + 42x - 9 \), we’ll start with the Rational Root Theorem. This theorem tells us that any rational root of the polynomial must be a factor of the constant term divided by a factor of the leading coefficient. So, we’ll consider the factors of -9 (constant term) and 14 (leading coefficient) to find potential rational roots. Upon checking possible roots such as \( \pm 1, \pm 3, \pm 9 \) divided by the factors of 14, we may find that \( x = \frac{1}{2} \) is a root. Next, we can apply synthetic division to factor \( f(x) \) and find additional roots, leading us eventually to discover all zeros and their respective multiplicities. As you dig deeper into polynomial equations, make sure to not overlook critical signs! Descartes' rule of signs can help you analyze the number of positive and negative roots quickly. You can count sign changes in the polynomial \( f(x) \) and \( f(-x) \) to guide your search for real roots. This approach saves time and narrows down the possible candidates. It’s a bit like being a detective in the world of functions!