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Factor Theorem

Find the valu...

Question

Find the value of a for which (x + 2a) is a factor of $(x^{5} - 4 a^{2} x^{3} + 2 x + 2 a + 3)$ .

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Solution

ANSWER:
Let $f (x) = x^{5} - 4 a^{2} x^{3} + 2 x + 2 a + 3$
It is given that (x + 2a) is a factor of f(x).
Using factor theorem, we have
f(−2a) = 0
⇒ $(- 2 a)^{5} - 4 a^{2} \times (- 2 a)^{3} + 2 \times - 2 a + 2 a + 3 = 0 \Rightarrow - 32 a^{5} - 4 a^{2} \times - 8 a^{3} + 2 \times - 2 a + 2 a + 3 = 0 \Rightarrow - 32 a^{5} + 32 a^{5} - 4 a + 2 a + 3 = 0 \Rightarrow - 2 a + 3 = 0 \Rightarrow 2 a = 3 \Rightarrow a = \frac{3}{2}$
Thus, the value of a is $\frac{3}{2}$

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