Find the value of a for which (x - 4) is a factor of $(2 x^{3} - 3 x^{2} - 18 x + a)$ .

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Solution

Let:
$f (x) = 2 x^{3} - 3 x^{2} - 18 x + a$
x-4 is a factor of $f (x) = 2 x^{3} - 3 x^{2} - 18 x + a \Rightarrow f (4) = 0 \Rightarrow 2 \times 4^{3} - 3 \times 4^{2} - 18 \times 4 + a = 0 \Rightarrow 8 + a = 0 \Rightarrow a = - 8$
Hence, the required value of a is -8.

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