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If the polynomials $ax^3 + 3x^2 - 13$ and $2x^3 - 5x + a$, when divided by $(x - 2)$, leave the same remainders, find the value of $a$.
Given:
The polynomials $ax^3 + 3x^2 - 13$ and $2x^3 - 5x + a$, when divided by $(x - 2)$, leave the same remainders.
To do:
We have to find the value of $a$.
Solution:
The remainder theorem states that when a polynomial, $p(x)$ is divided by a linear polynomial, $x - a$ the remainder of that division will be equivalent to $p(a)$.
Let $f(x) = ax^3 + 3x^2 - 13$ and $g(x) = 2x^3 - 5x + a$
$p(x) = x-2$
So, the remainders will be $f(2)$ and $g(2)$.
$f(2) = a(2)^3+3(2)^2 -13$
$= a(8) + 3(4) -13$
$=8a+12-13$
$=8a-1$
$g(2) = 2(2)^3-5(2) +a$
$= 2(8) -10 +a$
$=16-10+a$
$=a+6$
This implies,
$8a-1=a+6$
$8a-a=6+1$
$7a=7$
$a=1$
Therefore, the value of $a$ is $1$.
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