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Functions of polynomials often have more than one solution. For example, $x^2 = b$ with positive $b$ has two solutions for $x$.

How does that work for higher polynomials? Say, I have for positive $a,b,c$ and natural $y$

$$ (ax + b)^y = c \\ x_0 = \frac{c^{1/y}-b}{a}$$

Clearly, $x_0$ is one solution. How can I find potential real additional solutions for $y > 1$?

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  • $\begingroup$ Are you talking about real solutions, or are you also interested in complex solutions? $\endgroup$ – Pedro M. May 30 '15 at 0:18
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    $\begingroup$ Hint. Look up "fundamental theorem of algebra". The wikipedia page is a good place to start. $\endgroup$ – Ethan Bolker May 30 '15 at 0:18
  • $\begingroup$ Which are you interested in, finding the number of solutions or finding solutions? These are very different questions. $\endgroup$ – Robert Israel May 30 '15 at 0:25
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    $\begingroup$ @FooBar Ah. Sturm's theorem handles that (over the reals). $\endgroup$ – Milo Brandt May 30 '15 at 1:17
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    $\begingroup$ For real and the given example: if y is odd than one, if even than two. If you allow complex solution, than you have y. $\endgroup$ – Moti May 30 '15 at 6:06
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For real and the given example: if y is odd than one, if even than two. If you allow complex solution, than you have y

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