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If the polynomial $f(x)=\left|\begin{array}{ccc}(1+x)^{a} & (2+x)^{b} & 1 \\ 1 & (1+x)^{a} & (2+x)^{b} \\ (2+x)^{b} & 1 & (1+x)^{a}\end{array}\right|,$ then the
constant term of $f(x)$ is $[a$ and $b$ are positive integers $]$
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constant term of $f(x)$ is $[a$ and $b$ are positive integers $]$
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Verified Answer
The correct answer is:
$2-3 \cdot 2^{b}+2^{3 b}$
Given, $f(x)=\left|\begin{array}{ccc}(1+x)^{a} & (2+x)^{b} & 1 \\ 1 & (1+x)^{4} & (2+x)^{b} \\ (2+x)^{b} & 1 & (1+x)^{a}\end{array}\right|$
For constant term put $x=0$ $f(0)=\left|\begin{array}{ccc}1 & 2^{b} & 1 \\ 1 & 1 & 2 \\ 2^{b} & 1 & 1\end{array}\right|$
$=1\left(1-2^{b}\right)-2^{b}\left(1-2^{2 b}\right)+1\left(1-2^{b}\right)$
$=1-2^{b}-2^{b}+2^{3 b}+1-2^{b}$
$=2-3 \cdot 2^{b}+2^{3 b}$
For constant term put $x=0$ $f(0)=\left|\begin{array}{ccc}1 & 2^{b} & 1 \\ 1 & 1 & 2 \\ 2^{b} & 1 & 1\end{array}\right|$
$=1\left(1-2^{b}\right)-2^{b}\left(1-2^{2 b}\right)+1\left(1-2^{b}\right)$
$=1-2^{b}-2^{b}+2^{3 b}+1-2^{b}$
$=2-3 \cdot 2^{b}+2^{3 b}$
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