Question
If the polynomials $a x^3+4 x^2+3 x-4$ and $x^3-4 x+$ a leave the same remainder when divided by $(x-3)$, find the value of $a$.
✓
Answer
Let $p(x) = ax^3 + 4x^2+ 3x - 4$ and $q(x) = x^3 - 4x + a$ be the given polynomials.
When p(x) and q(x) are divided by (x - 3) the remainder are p(3) and q(3) respectively.
p(3) = q(3) given
$a(3)^3 + 4(3)^2 + 3 x 3 - 4 = 3^3- 4 x 3 + a$
$\Rightarrow 27a + 36 + 9 - 4 = 27 - 12 + a$
$\Rightarrow 26a = 15 - 41$
$\Rightarrow 26a = -26$
$\therefore a=-\frac{26}{26}$
$= -1.$
When p(x) and q(x) are divided by (x - 3) the remainder are p(3) and q(3) respectively.
p(3) = q(3) given
$a(3)^3 + 4(3)^2 + 3 x 3 - 4 = 3^3- 4 x 3 + a$
$\Rightarrow 27a + 36 + 9 - 4 = 27 - 12 + a$
$\Rightarrow 26a = 15 - 41$
$\Rightarrow 26a = -26$
$\therefore a=-\frac{26}{26}$
$= -1.$
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