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InterviewSolution
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Find rational roots of the polynomial f(x) = 2x3 + x2 - 7x - 6. |
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Answer» We have, f(x) = 2x 3+x 2-7x-6 Clearly, f (x) is a cubic polynomial with integer coefficients. if \(\frac{b}{c}\) is a rational root in lowest term, then the value of b are limited to the factors of 6 which are +1,+2,+3,+6 and values of c are limited to the factors of 2 which are +1,+2. Hence, the possible rational roots of f(x) are: +1,+2,+3,+6,+\(\frac{1}{2},\)+\(\frac{3}{2}\) We observe that, f (-1) = 2 (-1)3 + (-1)2 – 7 (-1) – 6 = -2 + 1 + 7 – 6 = 0 f (2) = 2 (2)3 + (2)2 – 7 (2) – 6 = 16 + 4 – 14 – 6 = 0 \(f(-\frac{3}{2})\) = 2 \((\frac{-3}{2})^{3}\) + \((\frac{-3}{2})^{2}\) - 7 \((\frac{-3}{2})-6\) = \(\frac{-27}{4}+\frac{9}{4}+\frac{21}{2}-6\) = 0 Hence, - 1, 2, \(\frac{-3}{2}\) are the rational roots of f (x). |
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