Question
$$14x^{16}+21x^{11}+7 \cdot \frac{x^6}{7}x^2 = 0$$
Answer
$$ x = 0 $$
Explanation
$$ \begin{aligned} 14x^{16}+21x^{11}+7 \cdot \frac{x^6}{7}x^2 &= 0&& \text{multiply ALL terms by } \color{blue}{ 7 }. \\[1 em]7\cdot14x^{16}+7\cdot21x^{11}+77 \cdot \frac{x^6}{7}x^2 &= 7\cdot0&& \text{cancel out the denominators} \\[1 em]98x^{16}+147x^{11}+7x^4 &= 0&& \\[1 em] \end{aligned} $$
In order to solve $ \color{blue}{ 98x^{16}+147x^{11}+7x^{4} = 0 } $, first we need to factor our $ x^4 $.
$$ 98x^{16}+147x^{11}+7x^{4} = x^4 \left( 98x^{12}+147x^{7}+7 \right) $$$ x = 0 $ is a root of multiplicity $ 4 $.
The remaining roots can be found by solving equation $ 98x^{12}+147x^{7}+7 = 0$.
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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