On dividing $x^3 - 3x^2 + x + 2$ by a polynomial $g(x)$, the quotient and remainder were $x - 2$ and $-2x + 4$, respectively. Find $g(x)$.

AcademicMathematicsNCERTClass 10

Given:

$x^3 - 3x^2 + x + 2$ is divided by $g(x)$.

The quotient and remainder were $x - 2$ and $-2x + 4$, respectively.

To do:

We have to find $g(x)$.

Solution:

Let $p(x) = x^3 – 3x^2 + x + 2$

Quotient $= x – 2$

Remainder $= -2x + 4$

On dividing $p(x)$ by $g(x)$, we have,

$p(x) = g(x) \times$ quotient $+$ remainder

$x^3– 3x^2 + x + 2 = g(x) (x – 2) + (-2x + 4)$

$x^3 – 3x^2 + x + 2 + 2 x- 4 = g(x) \times (x-2)$

$x^3 – 3x^2 + 3x – 2 = g(x) (x – 2)$

Therefore,

$g(x)=\frac{x^3 – 3x^2 + 3x – 2}{(x – 2)}$

$x-2$)$x^3-3x^2+3x-2$($x^2-x+1$

           $x^3-2x^2$

         ------------------------

           $-x^2+3x-2$

          $-x^2+2x$

         -------------------

                   $x-2$

                   $x-2$

               ------------

                   $0$

Therefore, $g(x)=x^2-x+1$

raja
Updated on 10-Oct-2022 13:19:38

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