Question

If the polynomial $f\left(x\right)=\left|\begin{array}{ccc}{(1+x)}^a& {(2+x)}^b& 1\\ 1& {(1+x)}^a& {(2+x)}^b\\ {(2+x)}^b& 1& {(1+x)}^a\end{array}\right|$, then the constant term of $f\left(x\right)$ is

• A. $2-3.2^b+2^{3b}$
• B. $2+3.2^b+2^{3b}$
• C. $2+3.2^b-2^{3b}$
• D. $2-3.2^b-2^{3b}$

Answer 1

The correct option is A $2-3.2^b+2^{3b}$

for constant term (put $x=0$)
$f\left(x\right)=\left|\begin{array}{ccc}1& 2^b& 1\\ 1& 1& 2^b\\ 2^b& 1& 1\end{array}\right|$

$=1(1-2^b)-2^b(1-2^b)+1(1-2^b)$

$=1-2^b-2^b+2^{3b}+1-2^b$

$=2-3.2^b+2^{3b}$