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- Vincent WongLv 71 十年前最愛解答
In mathematics, a discriminant is an expression that discriminates qualities of algebraic structures. The concept applies to polynomials, conic sections, quadratic forms, and algebraic number fields.
For a polynomial P(x) = a0 + a1x + a2x² + ... , the discriminant is a quantity D = D(a0,a1,a2,...) that equals 0 precisely for those P(x) that have at least one multiple root.
For a quadratic equation, the discriminant is the square-rooted section of the Quadratic Formula because you can use it to discriminate between whether the given quadratic has two solutions, one solution, or no solutions.
For instance, the quadratic polynomial P(x) = ax2 + bx + c has discriminant D = b2 − 4ac, which is the quantity under the square root sign in the quadratic formula. For real numbers a, b, c, one has:
When D > 0 , P(x) has two distinct real roots , and its graph crosses the x-axis twice.
When D = 0, P(x) has two coincided real roots , and its graph is tangent to the x-axis.
When D < 0 , P(x) has no real roots, and its graph lies strictly above or below the x-axis.
Discriminants in algebraic number theory are closely related, and contain information about ramification. In fact, the more geometric types of ramification are also related to more abstract types of discriminant, making this a central algebraic idea in many applications.
Discriminant of a polynomial
A discriminant of a polynomial is a number that can be easily computed from the coefficients of the polynomial and which is zero if and only if the polynomial has a multiple root. For instance, the discriminant of the polynomial ax2 + bx + c is b2 − 4ac.
For the general definition, suppose
is a polynomial with real coefficients. The discriminant of this polynomial is defined as the determinant of the (2n − 1)×(2n − 1) matrix
In the case n = 4, this discriminant looks like this:
The discriminant of p(x) is thus equal to the resultant of p(x) and p'(x), where p'(x) is the derivative of p(x).
One can show that, up to sign, the discriminant is equal to
where r1, ..., rn are the complex roots (counting multiplicity) of the polynomial p(x):
In fact, some authors define the discriminant by that formula, then show that the sign difference to the resultant is (−1)n(n −1)/2 .
It is clear from this second definition that, p has a multiple root if and only if the discriminant is zero. Note, however, that this multiple root can be complex.
In order to compute discriminants, one does not evaluate the above determinant each time for different coefficients, but instead evaluates it only once for general coefficients to get an easy-to-use formula. For instance, the discriminant of a polynomial of third degree p(x) = a3x3 + a2x2 + a1x + a0 is
The discriminant can be defined for polynomials over arbitrary fields, in exactly the same fashion as above. The product formula involving the roots ri remains valid; the roots have to be taken in some splitting field of the polynomial.
- 打倒美帝紙老虎Lv 61 十年前
- 1 十年前
- 1 十年前
a*x^2 + b*x + c = 0 為一元二次方程（quadratic equation)，
其解為 x = (-b +或- sqrt(b^2 - 4*a*c)/(2*a)；這裏有二個解(solutions)。
其中的 b^2 - 4*a*c 為該一元二次方程的判別式。我們設△=b^2 - 4*a*c。
△ 大於 0，二解為實數
△ 少於 0，二解為虛數
△ 等於 0，二解相同即重復
除此之外，判別式有一更廣義的義思，就是判別一代數結構的性質的表示式。例如：多項式P(x) = a0 + a1*x + a2*x^2 + ... + an*x^n = 0 亦有其相應的判別式，也是用來判別多項式的解的性質。