i.e., it discriminates the solutions of the equation (as equal and unequal real and nonreal) and hence the name 'discriminant'. It is helpful in determining what type of solutions a polynomial equation has without actually finding them. a = 1 a=1 a = 1 ) or changed by a substitution so that the summand of the second highest power vanishes. Discriminant of a polynomial in math is a function of the coefficients of the polynomial. The expressions become simpler if the polynomials are normalised (i.e.
( − 1 ) ⋅ a ⋅ Δ 2 = ∣ a b c 2 a b 0 0 2 a b ∣ = a b 2 + 4 a 2 c − 2 a b 2 (-1) \cdot a\cdot \Delta _2 = \begin The first n − 1 n-1 n − 1 lines of these matrices consist of the coefficients of the polynomial and the next n n n lines come from the coefficients of the 1st derivation of the polynomial: The terms can be obtained by calculating the determinants of the corresponding Sylvester matrices:
The discriminants for higher degree equations consist of an exponentially growing numbers of summands (4th degree: 16 5th degree: 59 6th degree: 246). It only has a single real solution if Δ = 0 \Delta =0 Δ = 0 and no real solution if Δ 0 \Delta >0 Δ > 0 then the equation has three distinct real solutions if Δ < 0 \Delta <0 Δ < 0 then the equation has one real and two conjugate complex solutions if Δ = 0 \Delta =0 Δ = 0 then all solutions are real, but at least two of them match. The criterion for quadratic equations is as follows:Ī quadratic equation a x 2 + b x + c = 0 ax^2+bx+c=0 a x 2 + b x + c = 0 has two real solutions, if for the coefficients a, b, c a, b, c a, b, c the discriminant Δ = b 2 − 4 a c > 0 \Delta = b^2-4ac >0 Δ = b 2 − 4 a c > 0.
The characterisation was done by an expression, for which he coined the term discriminant. In 1851, Sylvester discovered a criterion for cubic equations that allows statements to be made about the number and type of solutions.
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