How to do this polynomial question? (1 Viewer)

lolwot · Oct 31, 2012

Under the topic: roots and coefficients of polynomial equations

Two roots of

are equal and rational. Find m.

According to the textbook the answer is m = -9.

Thanks!

Carrotsticks · Dec 16, 2012

$\\ $Since two roots are equal, and we have a cubic (so it has 3 roots), we can let the roots be $ \alpha, \alpha, \beta $ where $ \alpha $ is rational as stated by the question.$ \\\\ \sum_{i} \alpha_i = 2 \alpha + \beta = -m \qquad (1) \\\\ \sum_{i<j} \alpha_i \alpha_j = \alpha \beta + \alpha \beta + \alpha ^2 = 2 \alpha \beta + \alpha ^2 = 15 \qquad (2) \\\\ \prod_{i} \alpha_i = \alpha ^2 \beta = 7 \qquad (3) \\\\ $From (3) we have $ \alpha \beta = \frac{7}{\alpha} $ and we substitute this into (2) to acquire $ \frac{14}{a} + \alpha ^2 = 15 \\\\ 14 + \alpha ^3 = 15 \alpha \\\\ \alpha ^3 - 15 \alpha + 14 = 0 \\\\ $By observation, we see that $ \alpha = 1 $ is a root and hence $ \alpha - 1 $ is a factor. Hence $ \\\\ \alpha ^3 - 14 \alpha + 15 = ( \alpha - 1)( \alpha ^2 + \alpha - 14) $ and the only rational (in fact, integer) root of this is $ \alpha 1 $ and therefore we also have $ \beta = 7 $ from (3).$ \\\\ $Therefore, $ m = - 2 \alpha - \beta = -2 -7 = -9$

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For the record, this question can be done in a couple of lines using the Multiple Root Theorem and/or the Rational (specifically, integer) Root Theorem, which is in the Extension 2 course.

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How to do this polynomial question? (1 Viewer)

lolwot

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Carrotsticks

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