Suppose a, b, c and x are real numbers. Let Δ$$=1+a1+ax1+ax21+b1+bx1+bx21+r1+rx1+rx2$$ .Then ∆ is independent of

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Suppose a, b, c and x are real numbers. Let Δ$$=1+a1+ax1+ax21+b1+bx1+bx21+r1+rx1+rx2$$ .Then ∆ is independent of

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Write Δ$$=$$Δ$$1+$$Δ$$2$$, where        Δ$$1=11+ax1+ax211+bx1+bx211+cx1+cx2and$$       Δ$$2=a$$    ax    $$ax2b$$    bx    $$bx2c$$    cx    $$cx2=0$$                 ∵$$C1$$ and $$C2$$ are propotional In Δ$$1$$,use $$C2$$→$$C2$$−$$C1$$,$$C3$$→$$C3$$−$$C1$$ to obtainΔ$$1=1axax21bxbx21cxcx2=0$$                     ∵$$C2$$ and $$C3$$ are propotional Thus Δ$$=0$$ and hence independence a,b,c,x.

Suppose $a, b, c$ and $x$ are real numbers. Let $Δ = |\begin{array}{ccc} 1 + a & 1 + a x & 1 + a x^{2} \\ 1 + b & 1 + b x & 1 + b x^{2} \\ 1 + r & 1 + r x & 1 + r x^{2} \end{array}|$ .Then $∆$ is independent of

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Suppose a, b, c and x are real numbers. Let Δ=1+a1+ax1+ax21+b1+bx1+bx21+r1+rx1+rx2 .Then ∆ is independent of

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Suppose $a, b, c$ and $x$ are real numbers. Let $Δ = |\begin{array}{ccc} 1 + a & 1 + a x & 1 + a x^{2} \\ 1 + b & 1 + b x & 1 + b x^{2} \\ 1 + r & 1 + r x & 1 + r x^{2} \end{array}|$ .Then $∆$ is independent of