Question
$$\frac{1+x}{2}+\frac{3-x}{4} = 17x-3x^6+11x^3$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} \frac{1+x}{2}+\frac{3-x}{4} &= 17x-3x^6+11x^3&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4 \cdot \frac{1+x}{2}+4\frac{3-x}{4} &= 4\cdot17x-4\cdot3x^6+4\cdot11x^3&& \text{cancel out the denominators} \\[1 em]2x+2+3-x &= 68x-12x^6+44x^3&& \text{simplify left and right hand side} \\[1 em]x+5 &= -12x^6+44x^3+68x&& \text{move all terms to the left hand side } \\[1 em]x+5+12x^6-44x^3-68x &= 0&& \text{simplify left side} \\[1 em]12x^6-44x^3-67x+5 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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