QuestionJuly 20, 2025

Factor the following polynomial. 91x^2+8x-3

Factor the following polynomial. 91x^2+8x-3
Factor the following polynomial. 91x^2+8x-3

Solution
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Answer

91(x - \frac{13}{91})(x + \frac{7}{31}) Explanation 1. Identify the coefficients Coefficients are a = 91, b = 8, c = -3. 2. Calculate the discriminant Discriminant \Delta = b^2 - 4ac = 8^2 - 4 \times 91 \times (-3) = 64 + 1092 = 1156. 3. Check if the discriminant is a perfect square \sqrt{1156} = 34, which is an integer, so it is a perfect square. 4. Use the quadratic formula to find roots Roots are given by x = \frac{-b \pm \sqrt{\Delta}}{2a}. x_1 = \frac{-8 + 34}{182} = \frac{26}{182} = \frac{13}{91}. x_2 = \frac{-8 - 34}{182} = \frac{-42}{182} = \frac{-21}{91} = \frac{-7}{31}. 5. Write the factorized form The polynomial factors as 91(x - \frac{13}{91})(x + \frac{7}{31}).

Explanation

1. Identify the coefficients<br /> Coefficients are $a = 91$, $b = 8$, $c = -3$.<br />2. Calculate the discriminant<br /> Discriminant $\Delta = b^2 - 4ac = 8^2 - 4 \times 91 \times (-3) = 64 + 1092 = 1156$.<br />3. Check if the discriminant is a perfect square<br /> $\sqrt{1156} = 34$, which is an integer, so it is a perfect square.<br />4. Use the quadratic formula to find roots<br /> Roots are given by $x = \frac{-b \pm \sqrt{\Delta}}{2a}$.<br /> $x_1 = \frac{-8 + 34}{182} = \frac{26}{182} = \frac{13}{91}$.<br /> $x_2 = \frac{-8 - 34}{182} = \frac{-42}{182} = \frac{-21}{91} = \frac{-7}{31}$.<br />5. Write the factorized form<br /> The polynomial factors as $91(x - \frac{13}{91})(x + \frac{7}{31})$.
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