Question

If (x¹⁰⁰ + 2x⁹⁹ + k) is divisible by (x + 1), then the value of k is
(a) 1
(b) 2
(c) −2
(d) −3

Answer 1

(a) 1

Let:
$p\left(x\right)=x^{100}+2x^{99}+k$
Now,
$x+1=0\Rightarrow x=-1$
$$\begin{gathered} \left(x+1\right)\text{is divisible by}p\left(x\right). \\ \therefore p\left(-1\right)=0 \\ \Rightarrow {\left(-1\right)}^{100}+2\times {\left(-1\right)}^{99}+k=0 \\ \Rightarrow 1-2+k=0 \\ \Rightarrow -1+k=0 \\ \Rightarrow k=1 \end{gathered}$$