Answer:
The Fundamental Theorem of Algebra (FTOA) tells us that any non-zero polynomial in one variable with complex (possibly real) coefficients has a complex zero.
A straightforward corollary, often stated as part of the FTOA is that a polynomial in a single variable of degree
n
>
0
with complex (possibly real) coefficients has exactly
n
complex (possibly real) zeros, counting multiplicity.
To see that the corollary follows, note that if
f
(
x
)
is a polynomial of degree
n
>
0
and
f
(
a
)
=
0
, then
(
x
−
a
)
is a factor of
f
(
x
)
and
f
(
x
)
x
−
a
is a polynomial of degree
n
−
1
. So repeatedly applying the FTOA, we find that
f
(
x
)
has exactly
n
complex zeros counting multiplicity.
Discriminants
If you want to know how many real roots a polynomial with real coefficients has, then you might like to look at the discriminant - especially if the polynomial is a quadratic or cubic. Ths discriminant gives less information for polynomials of higher degree.
The discriminant of a quadratic
a
x
2
+
b
x
+
c
is given by the formula:
Δ
=
b
2
−
4
a
c
Then:
Δ
>
0
indicates that the quadratic has two distinct real zeros.
Δ
=
0
indicates that the quadratic has one real zero of multiplicity two (i.e. a repeated zero).
Δ
<
0
indicates that the quadratic has no real zeros. It has a complex conjugate pair of non-real zeros.
The discriminant of a cubic
a
x
3
+
b
x
2
+
c
x
+
d
is given by the formula:
Δ
=
b
2
c
2
−
4
a
c
3
−
4
b
3
d
−
27
a
2
d
2
+
18
a
b
c
d
Then:
Δ
>
0
indicates that the cubic has three distinct real zeros.
Δ
=
0
indicates that the cubic has either one real zero of multiplicity
3
or one real zero of multiplicity
2
and another real zero.
Δ
<
0
indicates that the cubic has one real zero and a complex conjugate pair of non-real zeros.
Step-by-step explanation:
