Question

If $f\left(x\right)=\left|\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ a{x}^2& ax& a\end{array}\right|$ then $f\left(2x\right)-f\left(x\right)$ is divisible by

• A. $x$
• B. $a$
• C. $2a+3x$
• D. $x^2$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $x$
B $a$
C $2a+3x$

$f\left(2x\right)=\left[\begin{array}{ccc}a& -1& 0\\ 2ax& a& -1\\ 4a{x}^2& 2ax& a\end{array}\right]Applying,R_3-2x{R}_2,\left[\begin{array}{ccc}a& -1& 0\\ 2ax& a& -1\\ 0& 0& a+2x\end{array}\right]=(a+2x)(a^2+2ax)$
$f\left(x\right)=\left[\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ a{x}^2& ax& a\end{array}\right]Applying,R_3-x{R}_2,\left[\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ 0& 0& a+x\end{array}\right]=(a+x)(a^2+ax)$
Now,
$f\left(2x\right)-f\left(x\right)$
$=(a+2x)(a^2+2ax)-(a+x)(a^2+ax)$
$=a^3+2a^{2}x+2x{a}^2+4a{x}^2-a^3-a^{2}x-a^{2}x-a{x}^2$
$=3a{x}^2+2a^{2}x=ax(3x+2a)$

So, it is divisible by $a$, $x$ and ($3x+2a$)