Given \( f(x)=6 x^{3}+31 x^{2}+3 x-10 \), answer the following. Part: \( \mathbf{0} / 2 \) Part 1 of 2 Factor \( f(x) \), given that \( \frac{1}{2} \) is a zero. \( f(x)= \)
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To factor the polynomial \( f(x) = 6x^3 + 31x^2 + 3x - 10 \) given that \( \frac{1}{2} \) is a zero, we can use synthetic division. Using \( \frac{1}{2} \): \[ \begin{array}{r|rrrr} \frac{1}{2} & 6 & 31 & 3 & -10 \\ & & 3 & 17.5 & 10.75 \\ \hline & 6 & 34 & 20.5 & 0 \\ \end{array} \] The result is \( 6x^2 + 34x + 20.5 \). Now we can express it as: \[ f(x) = (x - \frac{1}{2})(6x^2 + 34x + 20.5) \] Next, we can continue to factor \( 6x^2 + 34x + 20.5 \). After some simplification and factoring out common factors, we find the fully factored form: \[ f(x) = (x - \frac{1}{2})(3x + 1)(2x + 41) \]