# Fundamental Theorem of Algebra

## Key Questions

See explanation...

#### Explanation:

Fundamental Theorem of Algebra

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 $f \left(x\right)$ is a polynomial of degree $n > 0$ and $f \left(a\right) = 0$, then $\left(x - a\right)$ is a factor of $f \left(x\right)$ and $f \frac{x}{x - a}$ is a polynomial of degree $n - 1$. So repeatedly applying the FTOA, we find that $f \left(x\right)$ has exactly $n$ complex zeros counting multiplicity.

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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:

$\Delta = {b}^{2} - 4 a c$

Then:

$\Delta > 0$ indicates that the quadratic has two distinct real zeros.

$\Delta = 0$ indicates that the quadratic has one real zero of multiplicity two (i.e. a repeated zero).

$\Delta < 0$ 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:

$\Delta = {b}^{2} {c}^{2} - 4 a {c}^{3} - 4 {b}^{3} d - 27 {a}^{2} {d}^{2} + 18 a b c d$

Then:

$\Delta > 0$ indicates that the cubic has three distinct real zeros.

$\Delta = 0$ indicates that the cubic has either one real zero of multiplicity $3$ or one real zero of multiplicity $2$ and another real zero.

$\Delta < 0$ indicates that the cubic has one real zero and a complex conjugate pair of non-real zeros.