Question

Brainleist question ,prove the statement in picture.

Answer 1

r2-r1 r3-r2 and simplify you get the answer

Answer 2

$\textbf{Hey there}$!!

$\textbf{Solution}$:


Taking L.H.S:
= $\begin{vmatrix}1&a&a^{2} \\ \\ 1&b&b^{2}\\ \\ 1&c&c^{2} \end{vmatrix}$

Applying $R_{1}$ ➡$R_{1}$ - $R_{2}$

= $\begin{vmatrix}0&a - b&a^{2} - b^{2} \\ \\ 1&b&b^{2}\\ \\ 1&c&c^{2} \end{vmatrix}$

Applying $R_{2}$ ➡$R_{2}$ - $R_{3}$

= $\begin{vmatrix}0&a - b&a^{2} - b^{2} \\ \\ 0&b - c&b^{2} - c^{2}\\ \\ 1&c&c^{2} \end{vmatrix}$

Applying $C_{1}$ ➡$C_{1}$ + $C_{2}$

= $\begin{vmatrix}a - b&a - b&(a - b)(a + b )\\ \\ b - c&b - c&(b - c)(b + c )\\ \\ 1 + c&c&c^{2} \end{vmatrix}$

Taking ( a - b )( b - c ) common from
$R_{1}$ and $R_{2}$ respectively.

= ( a - b )( b - c )$\begin{vmatrix}1&1&a + b \\ \\ 1&1&b + c\\ \\ 1 + c&c&c^{2} \end{vmatrix}$

Applying $C_{1}$ ➡$C_{1}$ - $C_{2}$

= ( a - b ) ( b - c )$\begin{vmatrix}0&1&a + b \\ \\ 0&1&b + c\\ \\ 1 &c&c^{2} \end{vmatrix}$

Expanding $C_{1}$

= ( a - b) (b - c ) 1[ b + c - a - b ]

= ( a - b ) ( b - c ) [b + c - a - b]

= ( a - b ) ( b - c ) ( c - a ) =R.H.S

#Be Brainly.