Wikipedia:Reference desk/Archives/Mathematics/2018 May 17

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May 17

Interpolating the roots of a polynomial from the roots of another

Given a root of the polynomial P is it possible to use that to find a root of polynomial Q if the two only differ by the last term? Like P=m^8-m^3+17 and Q=m^8-m^3-91. — Preceding unsigned comment added by 76.203.37.29 (talk) 13:55, 17 May 2018 (UTC)

It’s not possible in general. For example, Quintic function#Solvable quintics says $x^{5}-x-r=0$ has solutions in radicals if and only if it has an integer solution or r is one of ±15, ±22440, or ±2759640, in which cases the polynomial is reducible. So if P has one of those specific values of r and Q does not and has no integer root, then we can’t use the root of P to find a root of Q, at least not in radicals. Loraof (talk) 18:26, 17 May 2018 (UTC)

Does it make a difference if the roots are expressed in the form of approximations? Like x^5-x-53 as you say has no solution in radicals but it does have an complex root R at approximately (-1.78 - 1.28i). Could we not use R then to find a root of x^5-x+17? — Preceding unsigned comment added by 76.203.37.29 (talk) 21:04, 17 May 2018 (UTC)

The equation $x^{5}-x-53=0$ has the 5 solutions: 2.23, 0.67 ± 2.12i and -1.78 ± 1.28i , while $x^{5}-x+17=0$ has solutions: -1.80, -0.52 ± 1.70i, and 1.41 ± 1.00i . Solving the first equation did not help solving the second one. Bo Jacoby (talk) 04:44, 18 May 2018 (UTC).

I think I understand now. The article on Wilkinson's polynomial was helpful too. Thanks guys. — Preceding unsigned comment added by 76.203.37.29 (talk) 09:11, 18 May 2018 (UTC)