Difference between revisions of "Discriminant"
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====When <math>n=2</math>==== | ====When <math>n=2</math>==== | ||
− | Given a polynomial <math>p(x)=ax^2+bx+c</math>, its discriminant is <math>D(p)=b^2-4ac</math>, | + | Given a polynomial <math>p(x)=ax^2+bx+c</math>, its discriminant is <math>D(p)=b^2-4ac</math>, which can also be denoted by <math>\Delta=b^2-4ac</math>. |
For <math>\Delta>0</math> we have the graph | For <math>\Delta>0</math> we have the graph | ||
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[[Image:Delta_greater_than_0.png|thumb|center|300x300px]] | [[Image:Delta_greater_than_0.png|thumb|center|300x300px]] | ||
− | + | which has two distinct real roots. | |
For <math>\Delta<0</math> we have the graph | For <math>\Delta<0</math> we have the graph | ||
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[[File:Delta_lower_than_0.png|thumb|center|300x300px]] | [[File:Delta_lower_than_0.png|thumb|center|300x300px]] | ||
− | + | which has two non-real roots. | |
And for the case <math>\Delta=0</math>, | And for the case <math>\Delta=0</math>, | ||
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− | Also, the compressed [[cubic Equation|cubic]] form <math>p(z)=z^3+pz+q</math> has discriminant <math>D(p)=-4p^3-27q^2</math>. We can compress a polynomial of degree 3, | + | Also, the compressed [[cubic Equation|cubic]] form <math>p(z)=z^3+pz+q</math> has discriminant <math>D(p)=-4p^3-27q^2</math>. We can compress a polynomial of degree 3, which also makes possible to us to use Cardano's formula, by doing the substitution <math>x=z-\frac{a}{3}</math> on the polynomial <math>p(x)=x^3+ax^2+bx+c</math>. |
*If <math>D=0</math>, then at least two of the roots are equal; | *If <math>D=0</math>, then at least two of the roots are equal; |
Revision as of 19:19, 10 October 2023
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
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 ![$n=2$](//latex.artofproblemsolving.com/6/c/b/6cb7ae00764b56ee2adf59e78a1ffde9685b80db.png)
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 ![$n=3$](//latex.artofproblemsolving.com/5/6/3/5635737307f5f0651cced8ee2e6558a426fd27b5.png)
The discriminant of a polynomial is given by
.
Also, the compressed 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 ![$n=4$](//latex.artofproblemsolving.com/3/9/c/39cf3e35a4981583288d2a7c4b34989916fb7360.png)
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
.