If a- b- c- x- y- z - R- then a - x2 b - x2 c -x2 a - y2 b-y2 c - y2 a - z2 b -z2 c - z2-k - axp m - bxp n - cxp k - ayp m - by p n - cyp k - azp m - bzp n - czpFind the value of k-m-n-p -

(a x)^2 amp; (b x)^2 amp; (c x)^2 #160;(a y)^2 amp; #160;(b y)^2 amp; (c y)^2 (a z)^2 #160; amp; (b z)^2 amp; ...

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Factorisation by Regrouping Terms

If a, b, c, x...

Question

If a, b, c, x, y, z $\in$ R, then $∣ ∣ ∣ ∣ ∣ \begin{matrix} (a - x)^{2} & (b - x)^{2} & (c - x)^{2} (a - y)^{2} & (b - y)^{2} & (c - y)^{2} (a - z)^{2} & (b - z)^{2} & (c - z)^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} (k + a x)^{p} & (m + b x)^{p} & (n + c x)^{p} (k + a y)^{p} & (m + b y)^{p} & (n + c y)^{p} (k + a z)^{p} & (m + b z)^{p} & (n + c z)^{p} \end{matrix} ∣ ∣ ∣ ∣ ∣$
Find the value of $k + m + n + p$ ?

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∣ ∣ ∣ ∣ ∣ \begin{matrix} (a - x)^{2} & (a - y)^{2} & (a - z)^{2} (b - x)^{2} & (b - y)^{2} & (b - z)^{2} (c - x)^{2} & (c - y)^{2} & (c - z)^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 2 (b - c) (c - a) (a - b) (y - z) (z - x) (x - y)

a, b, c, x, y, z ϵ R

Δ_{1} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} {(a + x)}^{2} & {(a + y)}^{2} & {(a + z)}^{2} {(b + x)}^{2} & {(b + y)}^{2} & {(b + z)}^{2} {(c + x)}^{2} & {(c + y)}^{2} & {(c + z)}^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣

Δ_{2} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} {(1 - a x)}^{2} & {(1 - a y)}^{2} & {(1 - a z)}^{2} {(1 - b x)}^{2} & {(1 - b y)}^{2} & {(1 - b z)}^{2} {(1 - c x)}^{2} & {(1 - c y)}^{2} & {(1 - c z)}^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣

Δ_{1} - Δ_{2}

z : x :: z^{2} :

{|z|}^{2} > {|z|}^{2}

{|z|}^{2} = {|z|}^{2}

{|z|}^{2} < {|z|}^{2}

{|z|}^{2} \geq {|z|}^{2}

\frac{x^{2} (a - y) (a - z)}{(x - y) (x - z)} + \frac{y^{2} (a - z) (a - x)}{(y - z) (y - x)} + \frac{z^{2} (a - x) (a - y)}{(z - x) (z - y)}

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