Using factor theorem find the value of $a$ . If $2 x^{4} + a x^{3} + 4 x^{2} - x + 2$ is divisible by $2 x + 2$ .

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Solution

By factor theorm, if $x - a$ is a factor of polynomial $P (x)$ , then $P (a) = 0$ .

If ${2 x}^{2} + a x^{3} + {4 x}^{2} - x + 2$ is divisible by $2 x + 2$ then the polynomial must be zero at $x = - 1$

$∴ 2 {(- 1)}^{4} + a {(- 1)}^{3} + 4 {(- 1)}^{2} - (- 1) + 2 = 0$

$\Rightarrow 2 + 4 - a + 1 + 2 = 0$

$\Rightarrow a = 9$

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