Question

divide the polynomial f(x)=5x³+10x²-30x-15 by the polynomial g(x)=x²+1+x and hence, find the quotation and remainder​

Answer 1

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

Given polynomial is

$\rm :\longmapsto\:f(x) = {5x}^{3} + {10x}^{2} - 30x - 15$

Divisor is

$\rm :\longmapsto\:g(x) = {x}^{2} + x + 1$

So, using Long Division Method, we have

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\: 5x + 5\:\:}}}\\ {\underline{\sf{ {x}^{2} + x + 1 }}}& {\sf{\: {5x}^{3} + {10x}^{2} - 30x - 15 \:\:}} \\{\sf{}}& \underline{\sf{\:\:- 5{x}^{3} - 5{x}^{2} - 5x \: \: \: \: \: \: \: \: \: \: \: \: \: \:   \:  \:}} \\ {{\sf{}}}& {\sf{\: \:  \: \: \: \: \: \: 5{x}^{2} - 35x - 15  \:   }} \\{\sf{}}& \underline{\sf{\:\: \:  \:  \:  \: \: \: \: \: - 5{x}^{2}  -5x - 5  \:  \:  \:  \:  \:  \: \:\:}} \\ {\underline{\sf{}}}& {\sf{\: \:  \: \: \: \: \: \: \: \: \: \: \: \: - 40x - 20 \:\:}}   \end{array}\end{gathered}\end{gathered}\end{gathered}$

Thus,

$\rm :\longmapsto\:Quotient = 5x + 5$

and

$\rm :\longmapsto\:Remainder = - 40x - 20$

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Verification :-

Dividend is

$\rm :\longmapsto\:f(x)= {5x}^{3} + {10x}^{2} - 30x - 15$

Divisor is

$\rm :\longmapsto\:g(x) = {x}^{2} + x + 1$

Now, Consider

$\red{\rm :\longmapsto\:Divisor \times Quotient + Remainder}$

$\rm \:  =  \: (5x + 5)( {x}^{2} + x + 1) - 40x - 20$

$\rm \:  =  \: {5x}^{3} + {5x}^{2} + 5x + {5x}^{2} + 5x + 5 - 40x - 20$

$\rm \:  =  \: {5x}^{3} + {10x}^{2} - 30x - 15$

Hence, Verified