If the polynomial $6 x^{4} + 8 x^{3} - 5 x^{2} + a x + b$ is exactly divisible by the polynomial $2 x^{2} - 5$ , then find the value of $a$ and $b$ .

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Solution

Given polynomial
$p (x) = 6 x^{4} + 8 x^{3} - 5 x^{2} + a x + b$

And $g (x) = 2 x^{2} - 5$

So,

On divide $p (x)$ to $g (x)$ and we get,

Quotient $= 3 x^{2} + 4 x + 5$

Reminder $= (20 + a) x + (25 + b)$

Given that,

The polynomial $6 x^{4} + 8 x^{3} - 5 x^{2} + a x + b$ is exactly divisible by the polynomial $2 x^{2} - 5$ .

Hence,

$(20 + a) x + (25 + b) = 0$

$\Rightarrow 20 + a = 0$

$\Rightarrow a = - 20$

And $25 + b = 0$

$b = - 25$

Hence, this is the answer.

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