**Properties**

- The resultant is zero if and only if the two polynomials have a common root in an algebraically closed field containing the coefficients.
- Since the resultant is a polynomial with integer coefficients in term of the coefficients of
`P`and`Q`, it follows that- The resultant is well defined for polynomials over any commutative ring.
- If
`h`is a homomorphism of the ring of the coefficients into another commutative ring, which preserve the degrees of`P`and`Q`, then the resultant of the image by`h`of`P`and`Q`is the image by`h`of the resultant of`P`and`Q`.

- The resultant of two polynomials with coefficients in a integral domain is null if and only if they have a common divisor of positive degree.
- res(
`P`,`Q`) = (-1)deg`P`deg`Q`res(`Q`,`P`) - res(
`PR`,Q)=res(`P`,`Q`)res(`R`,`Q`) - If
`P`'=`P`+`RQ`and deg`P`'=deg`P`, then res(`P`,`Q`)=res(`P`',`Q`). - If
`X`,`Y`,`P`,`Q`have the same degree and`X`=`a`_{00}`P`+`a`_{01}`Q`,`Y`=`a`_{10}`P`+`a`_{11}`Q`,

- then

- res(
`P`_{-},`Q`)=res(`Q`_{-},`P`) where`P`_{-}(`z`)=`P`(-`z`)

Read more about this topic: Resultant

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