Question

If $p\left(x\right),q\left(x\right)$ and $r\left(x\right)$ are polynomials of degree one and $a_1,a_2$ and $a_3$ be real numbers, then $\Delta =\left|\begin{array}{ccc}p\left({\alpha }_1\right)& p\left({\alpha }_2\right)& p\left({\alpha }_3\right)\\ q\left({\alpha }_1\right)& q\left({\alpha }_2\right)& q\left({\alpha }_3\right)\\ r\left({\alpha }_1\right)& r\left({\alpha }_2\right)& r\left({\alpha }_3\right)\end{array}\right|=$

• A. $0$
• B. $2$
• C. $p\left({\alpha }_1\right)q\left({\alpha }_2\right)r\left({\alpha }_3\right)$
• D. None of theses

Answer 1

The correct option is A $0$

$p\left(x\right)=x+a$

$q\left(x\right)=x+b$

$r\left(x\right)=x+c$

$p\left({\alpha }_1\right)={\alpha }_1+ap\left({\alpha }_2\right)={\alpha }_2+ap\left({\alpha }_3\right)={\alpha }_3+a$

$q\left({\alpha }_1\right)={\alpha }_1+bq\left({\alpha }_2\right)={\alpha }_2+bq\left({\alpha }_3\right)={\alpha }_3+b$

$r\left({\alpha }_1\right)={\alpha }_1+cr\left({\alpha }_2\right)={\alpha }_2+cr\left({\alpha }_3\right)={\alpha }_3+c$

$\left|\begin{array}{ccc}{\alpha }_1+a& {\alpha }_2+a& {\alpha }_3+a\\ {\alpha }_1+b& {\alpha }_2+b& {\alpha }_3+b\\ {\alpha }_1+c& {\alpha }_2+c& {\alpha }_3+c\end{array}\right|$

$\Rightarrow \left|\begin{array}{ccc}{\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\alpha }_1& {\alpha }_2& {\alpha }_3\end{array}\right|+\left|\begin{array}{ccc}a& a& a\\ b& b& b\\ c& c& c\end{array}\right|$

$\Rightarrow {\alpha }_1{\alpha }_2\left|\begin{array}{ccc}1& 1& {\alpha }_3\\ 1& 1& {\alpha }_3\\ 1& 1& {\alpha }_3\end{array}\right|$

$△=0$

Option A