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Given \( f(x)=4 x^{3}+13 x^{2}-11 x-42 \), answer the following. Part: \( \mathbf{0} / \mathbf{2} \) Part 1 of 2 Factor \( f(x) \), given that -2 is a zero. \( f(x)= \)

Ask by O'Quinn Watson. in the United States
Mar 15,2025

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\( f(x) = (x + 2)(x + 3)(4x - 7) \)

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Beyond the Answer

To factor \( f(x) = 4x^3 + 13x^2 - 11x - 42 \) given that -2 is a zero, we can use synthetic division. Dividing by \( (x + 2) \) will give us the other factors. Performing synthetic division: \[ \begin{array}{r|rrrr} -2 & 4 & 13 & -11 & -42 \\ & & -8 & -10 & 42 \\ \hline & 4 & 5 & -21 & 0 \\ \end{array} \] The result is \( 4x^2 + 5x - 21 \). Now we can factor \( 4x^2 + 5x - 21 \) further. This can be factored using the quadratic formula or by finding two numbers that multiply to \( -84 \) (i.e., \( 4 \times -21 \)) and add to \( 5 \). The numbers \( 12 \) and \( -7 \) work. Thus, we write: \[ 4x^2 + 12x - 7x - 21 = (4x^2 + 12x) + (-7x - 21) = 4x(x + 3) - 7(x + 3) = (4x - 7)(x + 3) \] Therefore, the complete factorization of \( f(x) \) is: \[ f(x) = (x + 2)(4x - 7)(x + 3) \]

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