text: Find the remainder when $x^{3}+3 x^{2}+3 x+1$ is divided by(i) $x+1$(ii) $x-\frac{1}{2}$(iii) $x$(iv) $x+\pi$(v) $5+2 x$
  
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  To do:
  
  We have to find the remainder when $x^3+ 3x^2 + 3x + 1$ is divided by
  
  (i) $x+1$
  (ii) $x-\frac{1}{2}$
  (iii) $x$
  (iv) $x+\pi$
  (v) $5+2 x$
  
  Solution:
  
  The remainder theorem states that when a polynomial $p(x)$ is divided by a linear polynomial, $x - a$ the remainder of that division will be equivalent to $p(a)$.
  
  Let $f(x) =x^{3}+3 x^{2}+3 x+1$
  
  Therefore,
  
  (i) Let $p(x) = x +1$
  
  $=x-(-1)$
  
  So, the remainder will be $f(-1)$.
  
  $f(-1) =(-1)^{3}+3 (-1)^{2}+3 (-1)+1$
  
  $= -1+3(1)-3+1$
  
  $=-1+1+3-3$
  
  $=0$
  
  Therefore, the remainder is $0$.
  
  (ii) Let $q(x)=x-\frac{1}{2}$
  
  So, the remainder will be $f(\frac{1}{2})$.
  
  $f(\frac{1}{2}) =(\frac{1}{2})^{3}+3 (\frac{1}{2})^{2}+3 (\frac{1}{2})+1$
  
  $= \frac{1}{8}+3(\frac{1}{4})+\frac{3}{2}+1$
  
  $=\frac{1}{8}+\frac{3}{4}+\frac{3}{2}+1$
  
  $=\frac{1+3\times2+3\times4+1\times8}{8}$       (LCM of $8,4,2$ and $1$ is $8$)
  
  $=\frac{1+6+12+8}{8}$
  
  $=\frac{27}{8}$
  
  Therefore, the remainder is $\frac{27}{8}$.
  
  (iii) Let $r(x) = x$
  
  $=x-0$
  
  So, the remainder will be $f(0)$.
  
  $f(0) =(0)^{3}+3 (0)^{2}+3 (0)+1$
  
  $= 0+3(0)+0+1$
  
  $=1$
  
  Therefore, the remainder is $1$.
  
  (iv) Let $s(x) = x+\pi$
  
  $=x-(-\pi)$
  
  So, the remainder will be $f(-\pi)$.
  
  $f(-\pi) =(-\pi)^{3}+3 (-\pi)^{2}+3 (-\pi)+1$
  
  $= -\pi^3+3\pi^2-3\pi+1$
  
  Therefore, the remainder is $-\pi^3+3\pi^2-3\pi+1$.
  
  (v) Let $t(x) = 5+2x$
  
  $2x+5=0$
  
  $2x=-5$
  
  $x=\frac{-5}{2}$
  
  So, the remainder will be $f(-\frac{5}{2})$.
  
  $f(-\frac{5}{2}) =(-\frac{5}{2})^{3}+3 (-\frac{5}{2})^{2}+3 (-\frac{5}{2})+1$
  
  $= -\frac{125}{8}+3(\frac{25}{4})-\frac{15}{2}+1$
  
  $= -\frac{125}{8}+\frac{75}{4}-\frac{15}{2}+1$
  
  $=\frac{-125+75\times2-15\times4+1\times8}{8}$         (LCM of $8,4,2$ and $1$ is $8$)
  
  $=\frac{-125+150-60+8}{8}$
  
  $=\frac{158-185}{8}$
  
  $=\frac{-27}{8}$
  
  Therefore, the remainder is $-\frac{27}{8}$.
  
  Updated on 10-Oct-2022 13:39:07