First slide

Theory of equations

Question

$The value of k for which (a + 2 b) where a, b \neq 0 is a factor of a^{4} + 32 b^{4} + a^{3} b (k + 3) is$

Moderate

A

3
B

0
C

1
D

2

Solution

$\begin{array}{l} Let f (a) = a^{4} + 32 b^{4} + a^{3} b (k + 3) \\ f (- 2 b) = (- 2 b)^{4} + 32 b^{4} + (- 2 b)^{3} b (k + 3) = 0 \\ \Rightarrow 48 b^{4} - 8 b^{4} (k + 3) = 0 \\ \Rightarrow 8 b^{4} [6 - (k + 3)] = 0 \\ \Rightarrow 8 b^{4} (3 - k) = 0 \\ Since b \neq 0, 3 - k = 0 . \end{array}$

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