The quartic polynomial

\"\\displaystyle

\ \

has discriminant

\"\\Delta

\ \

\"-80abc^2de+18abcd^3+16ac^4e-4ac^3d^2-27b^4e^2+18b^3cde-4b^3d^3-4b^2c^3e+b^2c^2d^2.\"

\ \

Higher degrees \ \

More generally, for a polynomial of degree n with real coefficients, we have

Δ > 0: for some integer k such that $\"0$ , there are 2k pairs of complex conjugate roots and n-4k real roots, all different;
Δ < 0: for some integer k such that $\"0$ , there are 2k+1 pairs of complex conjugate roots and n-4k-2 real roots, all different;

Δ = 0: at least 2 roots coincide, which may be either real or not real (in this case their complex conjugates also coincide).

The polynomial equation is p(x) = 2x^4 - 7x^3 - 10x^2 + 21x - 12.

This is the fourth degree polynomial, so n = 4.

a = 2, b = - 7, c = - 10, d = 21 and e = - 12.

\"\\Delta

\ \

\"-80abc^2de+18abcd^3+16ac^4e-4ac^3d^2-27b^4e^2+18b^3cde-4b^3d^3-4b^2c^3e+b^2c^2d^2.\"

\ \

∆ = 256(8)(-1728) - 192(4)(-7)(21)(144) - 128(4)(100)(144) + 144(4)(-10)(441)(-12) - 27(4)(194481) + 144(2)(49)(-10)(-12) \ \

\ \

- 80(2)(-7)(100)(21)(-12) + 18(2)(-7)(-10)(9261) + 16(2)(10000)(-12) - 4(2)(-1000)(441) - 27(2401)(144) \ \

\ \

+ 18(-343)(-10)(21)(-12) - 4(-343)(-1000) + (49)(100)(441). \ \

\ \

∆ = - 3538944 + 2322432 - 7372800 + 30481920 - 21003948 + 1693440 - 28224000 + 23337720 - 3840000 + 3528000 \ \

\ \

- 9335088 - 15558480 - 1372000 + 2160900. \ \

\ \

∆ = - 26720848 < 0. \ \

\ \

Δ < 0: for some integer k such that $\"0$ , there are 2k+1 pairs of complex conjugate roots and n-4k-2 real roots, all different;