Question

2x³-x²-12 divided by x+3
The form of your answer should either be
p(x) or p(x)+k/x+3 where p(x) is a polynomial and k is an integer

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:\dfrac{ {2x}^{3} - {x}^{2} - 12}{x + 3}$

can be rewritten as

$\rm :\longmapsto\:\dfrac{ {2x}^{3} - {x}^{2} + 0x - 12}{x + 3}$

Now, using Method of Long Division, we have

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\: 2{x}^{2} -7x + 21\:\:}}}\\ {\underline{\sf{x + 3}}}& {\sf{\: 2{x}^{3}  -{x}^{2} + 0x - 12 \:\:}} \\{\sf{}}& \underline{\sf{\:  - 2{x}^{3} - 6{x}^{2}  \: \: \: \: \: \: \: \: \: \:   \:  \:  \:  \:  \: \:\:}} \\ {{\sf{}}}& {\sf{\: \:  \: \: \: \: \: \: -7{x}^{2} + 0x - 12 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: \: \: \: \: \: 7{x}^{2} + 21x  \:  \:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \:  \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 21x - 12  \:\:}} \\{\sf{}}& \underline{\sf{\: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: \: \: - 21x - 63\:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \:  \: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: \: \: \: \: \: \: \: \: \: \: - 75\:\:}}  \end{array}\end{gathered}\end{gathered}\end{gathered}$

Hence,

$\rm :\longmapsto\:\dfrac{ {2x}^{3} - {x}^{2} - 12}{x + 3} = {2x}^{2} - 7x + 21 + \dfrac{ - 75}{x + 3}$

VERIFICATION

Consider,

$\rm :\longmapsto\:{2x}^{2} - 7x + 21 + \dfrac{ - 75}{x + 3}$

$\rm \:  =  \: \dfrac{(x + 3)( {2x}^{2} - 7x + 21) - 75 }{x + 3}$

$\rm \:  =  \: \dfrac{{2x}^{3} - 7 {x}^{2} + 21x + {6x}^{2} - 21x + 63 - 75 }{x + 3}$

$\rm \:  =  \: \dfrac{ {2x}^{3} - {x}^{2} - 12}{x + 3}$

Hence, Verified