Show that 2x + 7 is a factor of 2x^3 + 5x^2 - 11 x - 14. Hence fa… - Vidyadip

Question

Show that $2x + 7$ is a factor of $2x^3 + 5x^2 - 11 x - 14.$ Hence factorise the given expression completely, using the factor theorem.

✓

Answer

If $2 x+7$ in factor of $2 x^3+5 x^2-11 x-14$ then on putting $2 x+7=0$
$x=-\frac{7}{2} $
$f\left(-\frac{7}{2}\right)=0 $
$=2\left(-\frac{7}{2}\right)^3+5\left(-\frac{7}{2}\right)^2-11\left(\frac{7}{2}\right)-14 $
$ =\frac{-343}{4}+\frac{245}{4}+\frac{77}{4}-14 $
$=\frac{-399}{4}+\frac{245+154}{4} $
$=\frac{-399+399}{4}=0$
Hence $2 x+7$ is one factor.
$\text { Now } 2 x^3+5 x^2-11 x-14 $
$=x^2(2 x+7)-x(2 x+7)-2(2 x+7)$
$=(2 x+7)\left(x^2-x-2\right)$
$=(2 x+7)\left(x^2+x-2 x-2\right) $
$=(2 x+7)[x(x+1)-2(x+1)] $
$=(2 x+7)(x-2)(x+1)$

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