User:Vqbc/Testing
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
, wich can also be denoted by
.
For we have the graph
wich has two distinct real roots.
For we have the graph
wich has two non-real roots.
And for the case ,
When 
The discriminant of a polynomial is given by
.
Also, the compressed cubic form has discriminant
. We can compress a polynomial of degree 3, wich 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
.