The polynomials $(a x^{3} + 3 x^{2} - 13)$ and $(2 x^{3} - 5 x + a)$ are divided by $(x + 2)$ . If the remainder in each case is the same, find the value of $a$ .

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Solution

Let $p (x) = a x^{3} + 3 x^{2} - 13$ and $g (x) = 2 x^{3} - 4 x + a$
By remainder theorem, the two remainders are $p (3)$ and $g (3)$
By the given condition, $p (3) = g (3)$
$∴ p (3) = a . 3^{3} + {3.3}^{2} - 13 = 27 a + 27 - 13 = 27 a + 14$
$g (3) = {2.3}^{3} - 4.3 + a = 54 - 12 + a = 42 + a$
Since $p (3) = g (3)$ , we get $27 a + 14 = 42 + a ∴ 26 a = 28$
$∴ a = \frac{28}{26} = \frac{14}{13}$

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