Find the value of a for which the polynomial $(x^{4} - x^{3} - 11 x^{2} - x + a)$ is divisible by (x +3).

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Solution

ANSWER:
Let:
$f (x) = x^{4} - x^{3} - 11 x^{2} - x + a$
Now,
x+3=0⇒x=-3
By the factor theorem, f(x) is exactly divisible by (x+3) if f(-3)=0.
Thus, we have:
$f (- 3) = (- 3)^{4} - (- 3)^{3} - 11 \times - 3^{2} - (- 3) + a = 81 + 27 - 99 + 3 + a = 12 + a$
Also,
$f (- 3) = 0 \Rightarrow 12 + a = 0 \Rightarrow a = - 12$
Hence, f(x) is exactly divisible by x+3 when a is -12

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