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When do polynomials have common roots? In my workbook is given such an exercise and, so , can you write please what's the condition for this thing to happen, so that two polynomials have one or more common roots. Thank you!

wonderingdev
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  • I'm not really sure what you're asking here. Can you be more specific? There are lots of different ways to analyze the roots of a polynomial. – Alex G. May 06 '14 at 12:48
  • For example I have two polynomials $P(X)=22x^4+33x^3-16ax^2-3x+2$ and $Q(x)=11x^4+33x^3+21x^2-2ax-2$. So it's required to find the values of real parameter $a$ such that the polynomials have common roots. – wonderingdev May 06 '14 at 12:50
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    You might find this helpful. – Marcin Łoś May 06 '14 at 13:02

2 Answers2

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Two polynomials have a common root if and only if their resultant vanishes.

The resultant $R(p(x), q(x))$ of two polynomials of degrees $m$ and $n$, respectively, is the determinant of the $(m + n) \times (m + n)$ matrix defined as follows. Write the coefficient of $p(x)$ in the first row followed by $n - 1$ zeros. In the next row the coefficients are displaced one place to the right, with one zero to the left and $n - 2$ zeros to the right. Continue in this fashion until the $nth$ row is $n - 1$ zeros followed by the coefficients of $p(x)$. For the last $m$ rows, we do the same thing with $p(x)$ and $q(x)$ interchanged.

With your example, we have the resultant as: $$ R(p(x), q(x)) = \mbox{det}\left(\begin{array}{cccccccc} 22 & 33 & -16a & -3 & 2 & 0 & 0 & 0\\ 0 & 22 & 33 & -16a & -3 & 2 & 0 & 0\\ 0 & 0 & 22 & 33 & -16a & -3 & 2 & 0\\ 0 & 0 & 0 & 22 & 33 & -16a & -3 & 2\\ 11 & 33 & 21 & -2a & -2 & 0 & 0 & 0\\ 0 & 11 & 33 & 21 & -2a & -2 & 0 & 0\\ 0 & 0 & 11 & 33 & 21 & -2a & -2 & 0 \\ 0 & 0 & 0 & 11 & 33 & 21 & -2a & -2 \end{array}\right).$$

Computer solution gives $a = \dfrac{3}{16}, \dfrac{297}{128},$ or $\dfrac{3}{2}$.

DYBnor
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You can perform the Euclidean Algorithm for finding the gcd of the two polynomials. They'll have common roots iff the gcd has degree at least 1.

This approach should work for the examples you gave in the comments.

lhf
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