Find the value of m for which (2x - 1) is a factor of $(8 x^{4} + 4 x^{3} - 16 x^{2} + 10 x + m)$ .

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Solution

ANSWER:
Let $f (x) = 8 x^{4} + 4 x^{3} - 16 x^{2} + 10 x + m$
It is given that (2x-1) is a factor of f(x).
Using factor theorem, we have
$f (\frac{1}{2}) = 0$
⇒ $8 \times {\frac{1}{2}}^{4} + 4 \times {\frac{1}{2}}^{3} - 16 \times {\frac{1}{2}}^{2} + 10 \times 12 + m = 0 \Rightarrow 12 + 12 - 4 + 5 + m = 0 \Rightarrow 2 + m = 0 \Rightarrow m = - 2$
Thus, the value of m is −2.

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