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1

There is a closed form expression for the roots of a quintic, it is just that the solutions do not always simplify to expressions involving field operations and radicals. For example, using Wolfram Alpha, one root of the equation $x^5-x+a=0$ is given by x_1 =a\phantom{.}{}_4F_3\left(\frac15, \frac25, \frac35, \frac45;\frac12, \frac34,\frac54;\frac{3125a^4}{...

2

We can rescale a polynomial so its leading coefficient is $1$ without changing the roots. So let's focus on polynomials with leading coefficient $1$. If its degree is $n$ then it has $n$ coefficients after the leading term. Writing down that polynomial amounts to saying what the $n$ non-leading coefficients are. When you are given a polynomial of degree $n$...

2

I advocate for an explanation of why the strategies for quadratics don't immediately work for higher orders. When finding the roots of a quadratic, there are two predominant strategies: factoring, and somehow taking its square root. Using Viete's formulas, we can get sum and product of roots. This is fine for quadratic since we have two roots and two ...

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