Find $a$ if the two polynomials $a x^{3} + 3 x^{2} - 9$ and $2 x^{3} + 4 x + a$ , leaves the same remainder when divided by $x + 3$ .

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Solution

Let $f (x) = a x^{3} + 3 x^{2} - 9$
and $g (x) = 2 x^{3} + 4 x + a$

putting $x + 3 = 0$
$x = - 3$

Since remainder is same $f (- 3) = g (- 3)$
$a {(- 3)}^{3} + 3 {(- 3)}^{2} - 9 = 2 {(- 3)}^{3} + 4 (- 3) + a$
$- 27 a + 27 - 9 = - 54 - 12 + a$
$27 a + a = 27 - 9 + 54 + 12$
$28 a = 84$
$a = 3$

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If the polynomials $a x^{3} + 3 x^{2}$ - 13 and $2 x^{3}$ - 5x + a, when divided by (x-2), leave the same remainders, find the value of a.

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g (x) = 2 x^{3} - 4 x + a

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a x^{3} + 3 x^{2} - 3

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