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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Updated on: 10-Oct-2022

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