If the polynomials $a x^{3} + 4 x^{2} + 3 x - 4 a n d x^{3} - 4 x + a$ leave the same remainder when divided by (x - 3), find the value of a.

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Solution

Let $p (x) = a x^{3} + 4 x^{2} + 3 x - 4 a n d q (x) = x^{3} - 4 x + a$ be the given polynomials. The remainders when p(x) and q(x) are divided by (x - 3) are p(3) and q(3) respectively.

By the given condition, we have
p(3) = q(3)
$\Rightarrow$ $a \times 3^{3} + 4 \times 3^{2} + 3 \times 3 - 4 = 3^{3} - 4 \times 3 + a$
$\Rightarrow$ $27 a + 36 + 9 - 4 = 27 - 12 + a$
$\Rightarrow$ $26 a + 26 = 0$
$\Rightarrow$ $26 a = - 26$
$\Rightarrow$ $a = - 1$

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