Without actual division prove that the polynomial $2x^{3} \ +\ 4x^{2} \ +\ x\ -\ 34\ =\ 0$ is exactly divisible by $(x\ -\ 2)$.

Given: $2x^{3} \ +\ 4x^{2} \ +\ x\ -\ 34\ =\ 0$ and $(x\ -\ 2)$.

To do: Here we have to check without using division method if $2x^{3} \ +\ 4x^{2} \ +\ x\ -\ 34\ =\ 0$ is exactly divisible by $(x\ -\ 2)$.

Solution:

If $x\ -\ 2$ is a factor, then $x\ =\ 2$ is a zero of $2x^{3} \ +\ 4x^{2} \ +\ x\ -\ 34\ =\ 0$.

Now,

$2x^{3} \ +\ 4x^{2} \ +\ x\ -\ 34\ =\ 0$

Putting $x\ =\ 2$ in the polynomial:

$2( 2)^{3} \ +\ 4( 2)^{2} \ +\ 2\ -\ 34\ =\ 0$

$2( 8) \ +\ 4( 4) \ -\ 32\ =\ 0$

$16\ +\ 16\ -\ 32\ =\ 0$

$32\ -\ 32\ =\ 0$

$\mathbf{0\ =\ 0}$

Therefore, the polynomial $2x^{3} \ +\ 4x^{2} \ +\ x\ -\ 34\ =\ 0$ is divisible by $x\ -\ 2$.

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Updated on: 10-Oct-2022

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