Question

Let $n>2$ be an integer and define a polynomial $p\left(x\right)=x^n+a_{n-1}{x}^{n-1}+.....+a_{1}x+a_0$ where $a_0,a_1,.....a_{n-1}$ are integers. Suppose we know that $np\left(x\right)=(1+x)p^{\prime }\left(x\right)$. If $b=p\left(1\right)$, then.

• A. B is divisible by $10$
• B. B is divisible by $3$
• C. B is a power of $2$
• D. B is a power of $5$

Answer 1

The correct option is C B is a power of $2$

Consider the differential equation,

$np\left(x\right)=(1+x)p^{\prime }\left(x\right)$

$\Rightarrow$ $ny=(1+x)\frac{dy}{dx}$

$\Rightarrow$ $\frac{ndx}{1+x}=\frac{dy}{y}$

Now on integrating on both sides

$\int \frac{ndy}{1+x}=\int \frac{dy}{y}$

$\Rightarrow$ $nln(1+x)+lnc=lny$

where $lnc$ is an arbitrary constant.

$\Rightarrow$ $y=p\left(x\right)=c(1+x{)}^n$

Now,substitute $x=0$

$\Rightarrow$ $c=a_0$

Now substitue $x=1$

Then, $b=p\left(1\right)=a_0{2}^n$

Therefore,Option C is correct.