The polynomial $f(x)=ax^3+bx-c$ is divisible by the polynomial $g(x)=x^2+bx+c$. Find the value of $ab$.
Given: The polynomial $f(x)=ax^3+bx-c$ is divisible by the polynomial $g(x)=x^2+bx+c$.
To do: To find the value of $ab$.
Solution:
As given, the polynomial $f(x)=ax^3+bx-c$ is divisible by the polynomial $g(x)=x^2+bx+c$.
By long division:
We find Remainder$=( ab^2+b-ac)x+( abc-c)
$\because f( x)$ is divisible by $g( x)$, remainder should be $0$.
$\Rightarrow ( ab^2+b-ac)x+( abc-c)=0$
$\Rightarrow ( ab^2+b-ac)x=0$ and $( abc-c)=0$
If $( abc-c)=0$
$\Rightarrow abc=c$
$\Rightarrow ab=1$
Thus, $ab=1$
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