Math, asked by snehaxa5, 5 hours ago

on dividing 3x^2 - 2x^2 + 5x + 5 by the polynomial p(x) , the quotient and remainder are x^2 - x + 2 and -7 respectively. find p(x)?​

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Answered by Vikramjeeth
6

*Question:-

on dividing 3x³ - 2x² + 5x - 5 by the polynomial p(x) , the quotient and remainder are x² - x + 2 and -7 respectively. find p(x)?

*Answer-

Kindly see the attachment for your answer

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Answered by mathdude500
2

Appropriate Question

On dividing 3x³ - 2x² + 5x - 5 by p(x), the quotient and remainder are x² - x + 2 and - 7 respectively. Find p(x).

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

Let assume that

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

Now

It is given that when f(x) is divided by p(x), the quotient q(x) and remainder r(x) is

\rm :\longmapsto\:q(x)  =  {x}^{2} - x + 2

and

\rm :\longmapsto\:r(x) =    - \: 7

We know,

By Euclid Division Algorithm,

\rm :\longmapsto\:f(x) = p(x) \times q(x) + r(x)

On substituting the values, we get

\rm :\longmapsto\:{3x}^{3} -  {2x}^{2} + 5x  -  5 = ( {x}^{2} - x + 2)q(x) - 7

\rm :\longmapsto\:{3x}^{3} -  {2x}^{2} + 5x  -  5  + 7= ( {x}^{2} - x + 2)q(x)

\rm :\longmapsto\:{3x}^{3} -  {2x}^{2} + 5x +2= ( {x}^{2} - x + 2)q(x)

\rm :\longmapsto\:q(x) = \dfrac{{3x}^{3} -  {2x}^{2} + 5x +2}{{x}^{2} - x + 2}

So, Using Long Division Method,

\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\: \:  \:  \:  \:  \:  \:  {3x} \:  +  \: 1\:  \:  \:  \: \:\:}}}\\ {{\sf{ {x}^{2} - x + 2}}}& {\sf{\: {3x}^{3} - {2x}^{2} +5x +2\:}} \\{\sf{}}&\underline{\sf{\:  -  {3x}^{3}  +  {3x}^{2}  -  6x \:  \:   \:\:  \:  \:  \:  \:  \:  \:  \:  \:   \:}}\\{\sf{}}&{\sf{\:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:\: {x}^{2}  - x + 2\:\:}}\\{\sf{}}&\underline{\sf{\:\: \:  \:  \:  \:  \:  \:  \:  \:   { - x}^{2}  + x - 2\:\:}}\\{\sf{}}&\underline{\sf{\: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:   \:  \:  \:  \:  \:  \:    \:   \: 0 \:  \:  \:  \:  \:  \:  \:  \: }}\end{array}\end{gathered}\end{gathered}\end{gathered}

Hence,

\bf\implies \:q(x) = 3x + 1

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