QuestionJuly 14, 2025

DO NOT USE A CALCULATOR Show all work for credit! A. Use synthetic division and/or factoring to write f(x)=28x^3+88x^2-37x-7 in completely factored form, given that (-(7)/(2)) is a zero of f(x) f(x)=square B. Find all the zeros of f(x) x=square

DO NOT USE A CALCULATOR Show all work for credit! A. Use synthetic division and/or factoring to write f(x)=28x^3+88x^2-37x-7 in completely factored form, given that (-(7)/(2)) is a zero of f(x) f(x)=square B. Find all the zeros of f(x) x=square
DO NOT USE A CALCULATOR Show all work for credit! A. Use synthetic division and/or factoring to write f(x)=28x^3+88x^2-37x-7 in completely factored form, given that (-(7)/(2)) is a zero of f(x) f(x)=square B. Find all the zeros of f(x) x=square

Solution
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Answer

A. f(x) = (x + \frac{7}{2})(x - \frac{1}{2})(x + \frac{1}{7}) ### B. x = -\frac{7}{2}, \frac{1}{2}, -\frac{1}{7} Explanation 1. Perform Synthetic Division Use -\frac{7}{2} as the divisor. Write coefficients: 28, 88, -37, -7. Bring down 28. Multiply by -\frac{7}{2} and add to next coefficient repeatedly. Calculation: - Bring down 28. - 28 \times -\frac{7}{2} = -98, add to 88: 88 - 98 = -10. - -10 \times -\frac{7}{2} = 35, add to -37: -37 + 35 = -2. - -2 \times -\frac{7}{2} = 7, add to -7: -7 + 7 = 0. 2. Write Quotient Polynomial Resulting polynomial from synthetic division is 28x^2 - 10x - 2. 3. Factor the Quadratic Factor 28x^2 - 10x - 2. Use the quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where a=28, b=-10, c=-2. Calculate discriminant: (-10)^2 - 4(28)(-2) = 100 + 224 = 324. Roots: x = \frac{10 \pm \sqrt{324}}{56} = \frac{10 \pm 18}{56}. Solutions: x = \frac{28}{56} = \frac{1}{2} and x = \frac{-8}{56} = -\frac{1}{7}. 4. Write Completely Factored Form Combine results: f(x) = (x + \frac{7}{2})(28x^2 - 10x - 2) = (x + \frac{7}{2})(x - \frac{1}{2})(x + \frac{1}{7}).

Explanation

1. Perform Synthetic Division<br /> Use $-\frac{7}{2}$ as the divisor. Write coefficients: 28, 88, -37, -7. Bring down 28. Multiply by $-\frac{7}{2}$ and add to next coefficient repeatedly.<br /> Calculation:<br />- Bring down 28.<br />- $28 \times -\frac{7}{2} = -98$, add to 88: $88 - 98 = -10$.<br />- $-10 \times -\frac{7}{2} = 35$, add to -37: $-37 + 35 = -2$.<br />- $-2 \times -\frac{7}{2} = 7$, add to -7: $-7 + 7 = 0$.<br /><br />2. Write Quotient Polynomial<br /> Resulting polynomial from synthetic division is $28x^2 - 10x - 2$.<br /><br />3. Factor the Quadratic<br /> Factor $28x^2 - 10x - 2$. Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=28$, $b=-10$, $c=-2$.<br /> Calculate discriminant: $(-10)^2 - 4(28)(-2) = 100 + 224 = 324$.<br /> Roots: $x = \frac{10 \pm \sqrt{324}}{56} = \frac{10 \pm 18}{56}$.<br /> Solutions: $x = \frac{28}{56} = \frac{1}{2}$ and $x = \frac{-8}{56} = -\frac{1}{7}$.<br /><br />4. Write Completely Factored Form<br /> Combine results: $f(x) = (x + \frac{7}{2})(28x^2 - 10x - 2) = (x + \frac{7}{2})(x - \frac{1}{2})(x + \frac{1}{7})$.
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