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V0ODO0CH1LD
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Homework Statement
Knowing that the equation:
[tex] X^n-px^2=q^m [/tex]
has three positive real roots a, b and c. Then what is
[tex] log_q[abc(a^2+b^2+c^2)^{a+b+c}] [/tex]
equal to?
Homework Equations
[tex] a + b + c = -(coefficient \ of \ second \ highest \ degree \ term) = -k_2 [/tex]
[tex] abc = -(constant \ coefficient) = k_4 [/tex]
[tex] a^2 + b^2 + c^2 = (coefficient \ of \ second \ highest \ degree \ term)^2-(coefficient \ of \ third \ highest \ degree \ term) = k^2_2-2k_3 [/tex]
The Attempt at a Solution
First I assumed X^3 is equal to k_1x^3 and so x^3 = X^3/k_1.
[tex] X^n-px^2=q^m\\
x^3-\frac{p}{k_1}x^2-\frac{q^m}{k_1}=0\\
k_2=-\frac{p}{k_1}\\
k_3=0\\
k_4=-\frac{q^m}{k_1}\\
a+b+c=\frac{p}{k_1}\\
abc=\frac{q^m}{k_1}\\
a^2+b^2+c^2=\frac{p^2}{k^2_1}\\
[/tex]
[tex] log_q[abc(a^2+b^2+c^2)^{a+b+c}]=\\
\frac{q^m}{k_1}(\frac{p^2}{k^2_1})^{\frac{p}{k_1}}=\\
log_q\frac{q^mp^{\frac{2p}{k_1}}}{k_1^{1+\frac{2p}{k_1}}}=\\
log_qq^mp^{\frac{2p}{k_1}}-log_qk_1^{1+\frac{2p}{k_1}}=\\
log_qq^m+log_qp^{\frac{2p}{k_1}}-log_qk_1^{1+\frac{2p}{k_1}}=\\
m+\frac{2p}{k_1}log_qp-(1+\frac{2p}{k_1})log_qk_1=\\
[/tex]
If I substitute (X^3)/(x^3) for k_1 I have an equation that only resembles the possible answers if X = x or X = -x, which makes me think that either X was a typo and they meant x. Or that somewhere in the problem statement they tell you about something that according to some property of polynomials makes X = x or X = -x implied.
Is there some weird property of polynomial that I don't know that would help me solve this problem? What is it?
EDIT: Sorry for the latex error..
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