Prove that: 1&a^2+bc&a^3 1&b^2+ca&b^3 1&c^2+ab&c^3 =-(a-b)(b-c)(c-b)(a^2+b^2+c^2)

1&a^2+bc&a^3 1&b^2+ca&b^3 1&c^2+ab&c^3 =-(a-b)(b-c)(c-b)(a^2+b^2+c^2) L.H.S= 1&a^2+bc&a^3 1&b^2+ca&b^3 1&c^2+ab&c^3 Apply: R_2 R_2-R_1 and R_3 R_3-R_1 = 1&a^2+bc&a^3 0&b^2+ca-a^2-bc&b^3-a^3 0&c^2+abb+ca-a^2-bc&c^3-a^3 = 1&a^2+bc&a^3 0&(b^2-a^2)-c(b-a)&b^3-a^3 0&(c^2-a^2)-b(c-a)&c^3-a^3 = 1&a^2+bc&a^3 0&(b-a)(b+a-c)&b^3-a^3 0&(c-a)(c+a-b)&c^3-a^3 =(b-a)(c-a) 1&a^2+bc&a^3 0&(b+a-c)&b^2+a^2+ab 0&(c+a-b)&c^2+a^2+ac =(b-a)(c-a) [(b+a-c)(c^2+a^2+ac) -(b^2+a^2+ab)(c^2+a^2+ac) ] =-(a-b)(b-c)(c-b)(a^2+b^2+c^2) =R.H.S

Prove that: 1&a^2+bc&a^3 1&b^2+ca&b^3 1&c^2+ab&c^3 =-(a-b)(b-c)(c…

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Question

Prove that:
$\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\1&\text{b}^2+\text{ca}&\text{b}^3\\1&\text{c}^2+\text{ab}&\text{c}^3 \end{vmatrix}$
$=-(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{b})(\text{a}^2+\text{b}^2+\text{c}^2)$

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Answer

$\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\1&\text{b}^2+\text{ca}&\text{b}^3\\1&\text{c}^2+\text{ab}&\text{c}^3 \end{vmatrix}$
$=-(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{b})(\text{a}^2+\text{b}^2+\text{c}^2)$
$\text{L.H.S}=\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\1&\text{b}^2+\text{ca}&\text{b}^3\\1&\text{c}^2+\text{ab}&\text{c}^3 \end{vmatrix}$
Apply: $R_2 \rightarrow R_2-R_1$ and $R_3 \rightarrow R_3-R_1$
$=\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\0&\text{b}^2+\text{ca}-\text{a}^2-\text{bc}&\text{b}^3-\text{a}^3\\0&\text{c}^2+\text{abb}+\text{ca}-\text{a}^2-\text{bc}&\text{c}^3-\text{a}^3 \end{vmatrix}$
$=\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\0&(\text{b}^2-\text{a}^2)-\text{c}(\text{b}-\text{a})&\text{b}^3-\text{a}^3\\0&(\text{c}^2-\text{a}^2)-\text{b}(\text{c}-\text{a})&\text{c}^3-\text{a}^3 \end{vmatrix}$
$=\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\0&(\text{b}-\text{a})(\text{b}+\text{a}-\text{c})&\text{b}^3-\text{a}^3\\0&(\text{c}-\text{a})(\text{c}+\text{a}-\text{b})&\text{c}^3-\text{a}^3 \end{vmatrix}$
$=(\text{b}-\text{a})(\text{c}-\text{a})\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\0&(\text{b}+\text{a}-\text{c})&\text{b}^2+\text{a}^2+\text{ab}\\0&(\text{c}+\text{a}-\text{b})&\text{c}^2+\text{a}^2+\text{ac} \end{vmatrix}$
$=(\text{b}-\text{a})(\text{c}-\text{a})\big[(\text{b}+\text{a}-\text{c})(\text{c}^2+\text{a}^2+\text{ac})\\-(\text{b}^2+\text{a}^2+\text{ab})(\text{c}^2+\text{a}^2+\text{ac})\big]$
$=-(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{b})(\text{a}^2+\text{b}^2+\text{c}^2)$
$=\text{R.H.S}$

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