I have a polynomial $P(x)$ that is already factored: $P(x) = (x+a_1)(x+a_2)...$. Is there a way to solve $P(x)=c$ where $c \neq 0$ ? Here the $a_i$ are all real. I'm guessing there isn't a general solution, but I'm hoping for a surprise.
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This boils down to determine the roots of a polynomial. Although it seems to have a very special form , I am pretty sure that we cannot easily solve it in general. – Peter Feb 10 '23 at 14:51
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Think about the graphs you've seen of polynomials of degree 3 or higher. The roots of the polynomial - the factors in your expression - are where it crosses the $x$-axis. But $y = c$ is a different horizontal line, and the solutions to your equation are where the graph crosses that line. Note that if you raise or lower that line, some crossings move closer together, and some move farther apart, and as the curve has different slopes on each side of the hills and valleys, the rate of movement of those crossings are different. They are not simple to track. But we still have Newton's method. – Paul Sinclair Feb 11 '23 at 17:00