Check whether $2x + 3$ is the factor of $2x^3 + x^2 - 5x + 2$.

Given :

$f(x) = 2x^3 + x^2 - 5x + 2$, $g(x) = 2x+3$ are the given polynomials.

To do :

We have to check whether $2x + 3$ is the factor of $2x^3 + x^2 - 5x + 2$.

Solution :

$2x +3 = 0$

$2x = -3$

$x = \frac{-3}{2}$

If g(x) is a factor of f(x) then  $\frac{-3}{2}= 0$.

 $f(\frac{-3}{2}) = 2(\frac{-3}{2})^3+ (\frac{-3}{2})^2 - 5 (\frac{-3}{2}) + 2$

             $= -2 \frac{27}{8} + \frac{9}{4} + \frac{15}{2} +2$

             $= \frac{-27}{4} + \frac{9}{4} + \frac{15}{2} + 2$

             $= \frac{(-27+9+15\times2+2\times4)}{4}$

            $ = \frac{(-18+30+8)}{4}$

             $=\frac{20}{4}$

             $= 5$

 $f(\frac{-3}{2})$ is not equal to zero.

Therefore,

$2x+3$ is not a factor of  $2x^3 + x^2 - 5x + 2$.

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Updated on: 10-Oct-2022

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