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Find d delta(discriminant) of 2x^3-7x^2-17x+10
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#thanks
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.
To find the delta (discriminant) of a polynomial, you first need to determine the coefficients of the polynomial equation. The equation you provided is:
2x^3 - 7x^2 - 17x + 10
Step 1: Identify the coefficients of the polynomial equation:
The coefficient of x^3 is 2
The coefficient of x^2 is -7
The coefficient of x is -17
The constant term is 10
Step 2: Calculate the discriminant:
The discriminant of a cubic polynomial is given by the formula:
delta = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd
In this case, the coefficients are:
a = 2 (the coefficient of x^3)
b = -7 (the coefficient of x^2)
c = -17 (the coefficient of x)
d = 10 (the constant term)
Now, substitute these values into the formula and calculate the discriminant:
delta = (-7)^2 * (-17)^2 - 4 * 2 * (-17)^3 - 4 * (-7)^3 * 10 - 27 * 2^2 * 10^2 + 18 * 2 * (-7) * (-17) * 10
Simplifying this expression will give you the value of the delta (discriminant).