according to the fundamental theorem of algebra how many roots exits for the polynomial function. f (x) = 8x^7-x^5+x^3+6

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.