Question
$$\frac{x^9+x^5}{x^8+x^6} = \frac{7}{3}$$
Answer
$$ x = 0 $$
Explanation
$$ \begin{aligned} \frac{x^9+x^5}{x^8+x^6} &= \frac{7}{3}&& \text{multiply ALL terms by } \color{blue}{ (x^8+x^6)\cdot3 }. \\[1 em](x^8+x^6)\cdot3 \cdot \frac{x^9+x^5}{x^8+x^6} &= (x^8+x^6)\cdot3\cdot\frac{7}{3}&& \text{cancel out the denominators} \\[1 em]3x^9+3x^5 &= 7x^8+7x^6&& \text{move all terms to the left hand side } \\[1 em]3x^9+3x^5-7x^8-7x^6 &= 0&& \text{simplify left side} \\[1 em]3x^9-7x^8-7x^6+3x^5 &= 0&& \\[1 em] \end{aligned} $$
In order to solve $ \color{blue}{ 3x^{9}-7x^{8}-7x^{6}+3x^{5} = 0 } $, first we need to factor our $ x^5 $.
$$ 3x^{9}-7x^{8}-7x^{6}+3x^{5} = x^5 \left( 3x^{4}-7x^{3}-7x+3 \right) $$$ x = 0 $ is a root of multiplicity $ 5 $.
The remaining roots can be found by solving equation $ 3x^{4}-7x^{3}-7x+3 = 0$.
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using quartic formulas
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