Question

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Find d delta(discriminant) of 2x^3-7x^2-17x+10

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#thanks

Answer 1

hmmm. working with cubics? Gets messy. But, you should be able to find out that the discriminant is

Δ = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd

For your polynomial, that's just 99225

As wikipedia explains,

For a cubic polynomial with real coefficients, the discriminant reflects the nature of the roots as follows:

Δ > 0: the equation has 3 distinct real roots;
Δ < 0, the equation has 1 real root and 2 complex conjugate roots;
Δ = 0: at least 2 roots coincide, and they are all real.

It may be that the equation has a double real root and another distinct single real root; alternatively, all three roots coincide yielding a triple real root.

If a cubic polynomial has a triple root, it is a root of its derivative and of its second derivative, which is linear. Thus to decide if a cubic polynomial has a triple root or not, one may compute the root of the second derivative and look if it is a root of the cubic and of its derivative.