Discriminant
The discriminant of a quadratic equation of the form is the quantity . When are real, this is a notable quantity, because if the discriminant is positive, the equation has two real roots; if the discriminant is negative, the equation has two nonreal roots; and if the discriminant is 0, the equation has a real double root.
Contents
[hide]Discriminant of polynomials of degree n
The discriminant can tell us something about the roots of a given polynomial of degree with all the coefficients being real. But for polynomials of degree 4 or higher it can be difficult to use it.
General formula of discriminant
We know that the discriminant of a polynomial is the product of the squares of the differences of the polynomial roots , so,
When
Given a polynomial , its discriminant is , which can also be denoted by .
For we have the graph
which has two distinct real roots.
For we have the graph
which has two non-real roots.
And for the case ,
When
The discriminant of a polynomial is given by .
Also, the depressed cubic form has discriminant . We can compress a polynomial of degree 3, which also makes possible to us to use Cardano's formula, by doing the substitution on the polynomial .
- If , then at least two of the roots are equal;
- If , then all three roots are real and distinct;
- If , then one of the roots is real and the other two are complex conjugate.
When
The quartic polynomial has discriminant
- If , then at least two of the roots are equal;
- If , then the roots are all real or all non-real;
- If , then there are two real roots and two complex conjugate roots.
Some properties
For we can say that
- The polynomial has a multiple root if, and only if, ;
- If , with being a positive integer such that , with being the degree of the polynomial, then there are pairs of complex conjugate roots and real roots;
- If , with being a positive integer such that , then there are pairs of complex conjugate roots and real roots.
Example Problems
Introductory
- (AMC 12 2005) There are two values of for which the equation has only one solution for . What is the sum of these values of ?
Solution: Since we want the 's where there is only one solution for , the discriminant has to be . . The sum of these values of is .