Question

If $\Delta =\left|\begin{array}{ccc}l& a^2& a^3\\ l& b^2& b^3\\ l& c^2& c^3\end{array}\right|$, then $\Delta$ is divisible by

• A. $a-b$
• B. $b-c$
• C. $(c-a)(ab+bc+ca)$
• D. All of these

Answer 1

The correct option is D All of these

$\Delta =\left|\begin{array}{ccc}1& a^2& a^3\\ 1& b^2& b^3\\ 1& c^2& c^3\end{array}\right|$
$R_1\to R_1-R_2,R_2\to R_2-R_3$
$\Delta =\left|\begin{array}{ccc}0& a^2-b^2& a^3-b^3\\ 0& b^2-c^2& b^3-c^3\\ 1& c^2& c^3\end{array}\right|$
$\Delta =(a-b)(b-c)\left|\begin{array}{ccc}0& a+b& a^2+ab+b^2\\ 0& b+c& b^2+bc+c^2\\ 1& c^2& c^3\end{array}\right|$
$=(a-b)(b-c)\left[\right(a+b\left)\right(b^2+bc+c^2)-(b+c\left)\right(a^2+ab+b^2\left)\right]$
$=(a-b)(b-c)[a{c}^2-a^{2}c+b{c}^2-a^{2}b]$
$=(a-b)(b-c)(c-a)(ab+bc+ac)$
Hence, all options are correct