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how can I show that the equation $x^5+5px^3+5p^2x+q=0$ will have a pair of equal roots, if $q^2+4p^5=0$.

can anyone help me.thanks a lot.

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2 Answers

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A polynomial has a pair of equal roots if it has a common factor with its derivative, so you might want to compute the derivative and then see whether that condition gives it a common root with the original polynomial.

There is a more systematic approach, involving the resultant (which you can look up). The polynomial and its derivative have a common root if and only if the resultant of the polynomial and its derivative is zero. So, you can calculate the resultant, then see whether $q^2+4p^5$ is a factor of the resultant.

These seem like unlikely approaches for something at the level of algebara-precalculus, though.

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The discriminant vanishes iff there are common roots of the polynomial. In this case the discriminant is $3125(4p^2+q^2)^2$.

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