Question

if x^11 is divided by x+1 what is the remainder?​

Answer 1

Answer:

g(x) = x+ 1

      = x+1 = 0

      = x = - 1

p(x) = x^11

p( - 1) = (-1)^11

         =  -1

Answer 2

Here we recall the so called "Remainder Theorem"!

$\textit{``The polynomial p(x) leaves remainder p(a) on division by (x-a). "}$

Proof for remainder theorem is given below.

Let the polynomial $p(x)$ on division by $(x-a)$ gives quotient $q(x)$ and remainder $r(x).$ Thus, by Euclid's Division Algorithm, we have,

$\longrightarrow p(x)=q(x)\cdot(x-a)+r(x)$

Consider $x=a,$ since $x$ is able to accept all possible real number values. Then,

$\longrightarrow p(a)=q(a)\cdot(a-a)+r(a)$

$\longrightarrow p(a)=q(a)\cdot0+r(a)$

$\longrightarrow p(a)=r(a)$

This implies $p(a)$ itself is the remainder. Hence the Proof.

Here, according to the question,

$\longrightarrow p(x)=x^{11}$

And,

$\longrightarrow x-a=x+1$

$\Longrightarrow a=-1$

Hence the remainder obtained on dividing $x^{11}$ by $(x+1)$ is,

$\longrightarrow p(a)=(-1)^{11}$

$\longrightarrow\underline{\underline{p(a)=-1}}$

Hence -1 is the answer.