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I have a polynomial $p(x) = \prod_{j=1}^m (x-a_j)$. Now I make a new polynomial by adding a constant, $q(x) = p(x) + c$. Can I use my knowledge of the roots of $p(x)$ to make it easier to find the roots of $q(x)$?

Clarifications: The constants are real and I'm generally only interested in real roots. I apologize for the vague use of "easier". One definition would be be that you can write the new roots in terms of the old ones using elementary operations. Milo Brandt has shown that this not possible. On the other hand, in the comments, Travis points out that if $c$ is small, then the old roots will be good guesses for finding the new roots. With this in mind, I will define "easier" as follows. I could expand $q(x)$ and input it into a generic numerical root finding algorithm; is there a way to use knowledge of $p(x)$ to find roots faster or more accurately?

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Greg Ver Steeg
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  • It depends on what you mean, but if $c$ is small in a suitable sense, the original roots $a_j$ are good candidates for seed values when applying Newton's Method to approximate the roots of $q(x)$. – Travis Willse Jan 07 '16 at 00:34
  • Thanks! That's a good point. Sadly $c$ is large in my case. – Greg Ver Steeg Jan 07 '16 at 00:34
  • If $p$ is real, when $|c|$ is large enough (or sometimes even when it is small when some root $a_j$ is repeated an even number of times) then $q$ will in general have fewer than $\deg q = m$ real roots. One can use the $a_j$ and $c$ to extract some conditions on the roots of $q$ using Rouche's Theorem (https://en.wikipedia.org/wiki/Rouch%C3%A9%27s_theorem), but whether this is useful depends on what one means by "easier to find the roots". (In fact, this works for a general complex polynomial $\Bbb C$, in particular a real polynomial with some complex roots.) – Travis Willse Jan 07 '16 at 00:42
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    If you are seeking real roots, then for $|c|$ large enough, $p(x) + c$ will have one or two or no roots! – Unit Jan 07 '16 at 00:45
  • This is another great point. In fact, we can easily calculate the sign over each interval for $p(x)$. If $c$ is positive, we can rule out certain intervals as potential roots. – Greg Ver Steeg Jan 07 '16 at 01:31

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There's no way to find the roots of a shift of a polynomial from the roots of the polynomials itself using only radicals, addition, multiplication, and division. In particular, adding a constant to a polynomial, one can change a nice well behaved polynomial like: $$x^5-x=0$$ which factors as $$x(x-1)(x+1)(x-i)(x+i)$$ into a polynomial which has no solutions expressible in terms of radicals: $$x^5-x+1=0.$$ This is a problem, as it rules out there being any "easy" formula relating the roots of a polynomial to the roots of a translate.

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