How can I count all the zeros of polynomials by using discriminant?

The discriminant of a polynomial can be used to determine the number of real and complex zeros (roots) of the polynomial. For a polynomial of degree n, the discriminant can provide valuable information about its zeros. Here's how you can use the discriminant to count the zeros:

For a quadratic polynomial (degree 2): Consider a quadratic polynomial of the form ax^2 + bx + c = 0. The discriminant (D) of this polynomial is given by:

D = b^2 - 4ac

Now, depending on the value of the discriminant (D), we can determine the number and nature of the zeros:

a. If D > 0, the polynomial has two distinct real roots. b. If D = 0, the polynomial has two real roots that are equal (a double root). c. If D < 0, the polynomial has two complex (conjugate) roots.

For a cubic polynomial (degree 3): Consider a cubic polynomial of the form ax^3 + bx^2 + cx + d = 0. The discriminant (D) of this polynomial is given by:

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

Depending on the value of the discriminant (D), we can determine the number and nature of the zeros:

a. If D > 0, the polynomial has three distinct real roots. b. If D = 0, the polynomial may have three real roots, but at least two of them are equal (at least one double root). c. If D < 0, the polynomial has one real root and two complex (conjugate) roots.

For a quartic polynomial (degree 4): Consider a quartic polynomial of the form ax^4 + bx^3 + cx^2 + dx + e = 0. The discriminant (D) of this polynomial is given by:

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

Depending on the value of the discriminant (D), we can determine the number and nature of the zeros:

a. If D > 0, the polynomial has four distinct real roots. b. If D = 0, the polynomial may have four real roots, but at least two of them are equal (at least one double root). c. If D < 0, the polynomial may have two real roots and two complex (conjugate) roots.

It's important to note that while the discriminant can give us information about the number and nature of the zeros, it does not provide the actual values of the roots. To find the roots themselves, you may need to use numerical methods or other techniques, depending on the degree of the polynomial and its complexity.

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