Question

divide x^4+1 by x-1 and find quotient and remainder

Answer 1

Divide x^4+1 by x-1 and find quotient and remainder

$\underline{\textbf{Given:}}$

$\mathsf{x^4+1{\div}x-1}$

$\underline{\textbf{To find:}}$

$\mathsf{The\;qutient\;and\;remainder\;when\;x^4+1}$

$\mathsf{is\;divided\;by\;(x-1)}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{\dfrac{x^4+1}{x-1}}$

$\textsf{This can be written as,}$

$\mathsf{=\dfrac{(x^4-1)+2}{x-1}}$

$\mathsf{=\dfrac{((x^2)^2-1^2)+2}{x-1}}$

$\textsf{Using the identity,}\;\boxed{\mathsf{a^2-b^2=(a-b)(a+b)}}$

$\mathsf{=\dfrac{(x^2-1)(x^2+1)+2}{x-1}}$

$\mathsf{=\dfrac{(x^2-1^2)(x^2+1)+2}{x-1}}$

$\mathsf{=\dfrac{(x-1)(x+1)(x^2+1)+2}{x-1}}$

$\mathsf{=\dfrac{(x-1)(x+1)(x^2+1)}{x-1}+\dfrac{2}{x-1}}$

$\mathsf{=(x+1)(x^2+1)+\dfrac{2}{x-1}}$

$\mathsf{=x^3+x^2+x+1+\dfrac{2}{x-1}}$

$\textsf{On division, we get}$

$\mathsf{Quotient=x^3+x^2+x+1}$

$\mathsf{Remainder=2}$

Answer 2

Answer:

remainder=2, quotient=x^3+x^2+x+1

Step-by-step explanation:

see the image below