Question

Find the remainder when x^51+51 is divided by x+1

Answer 1

${\huge{\green{\underline{\underline{\sf {\mathfrak{Answer}}}}}}}$

${\bf{\orange{\underline{Given:}}}}$

• $p(x)=x^{51}+51$

${\bf{\orange{\underline{Solution:}}}}$

$\bf{\pink{\underline{\bigstar{Let:}}}}$

$\implies x + 1 = 0$

$\implies x = - 1$

$\bf{\pink{\underline{\bigstar{Finding \ p(-1):}}}}$

$\implies p(-1)= (-1)^{51} +51 \\ \implies p(-1) = -1 + 51\\ \implies p(-1)=50$

$\green{\boxed{\implies{\boxed{Remainder = 50}}}}$

Answer 2

Answer -

» Given:

Quotient: $x^{51} +51$

Divisor = x+1

» To find:

remainder .

» We know that,

${\pink{\boxed{Remainder \: theorem}}}$ -

Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a).

» By applying remainder theorem -

x +1 = 0

x = -1

$x^{51}+51 \\ \\ \\ p(-1) = (-1)^{51} +51\\ \\ \\ = -1+51 = 50\\ \\ \\ {\pink{\boxed{therefore\:remainder \:when\: p(x) \:is\:divided\:by\:(x+1)= 50}}}$