1.1.3 Polynomial functions

FE Mechanical

1. Mathematics

1.1. Analytic geometry

Our FE Mechanical course is currently in development and is a work-in-progress.

Polynomial functions

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A polynomial function is a function of the form:

$P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0},$

where:

$a_{n}, a_{n - 1}, \dots, a_{1}, a_{0}$ are constants called coefficients,
$n$ is a non-negative integer called the degree of the polynomial, and
$a_{n} \neq = 0$ (the leading coefficient).

The fundamental theorem of algebra

The fundamental theorem of algebra is one of the most important results in mathematics:

Every polynomial of degree $n \geq 1$ with complex coefficients has exactly $n$ roots (counting multiplicities) in the complex numbers.

Its key implications include:

Existence of roots. Every non-constant polynomial has at least one root.
Complete factorization. Any polynomial can be completely factored into linear factors over the complex numbers.
Counting roots. A polynomial of degree $n$ has exactly $n$ roots when multiplicities are counted.

Polynomials with real coefficients

When a polynomial has real coefficients, which is the case in most applications in science and engineering, its roots have special properties:

Complex conjugate root theorem

If a polynomial with real coefficients has a complex root $a + bi$ (where $b \neq = 0$ ), then its complex conjugate $a - bi$ is also a root.

The consequences of this theorem include:

Complex roots come in pairs for polynomials with real coefficients.
Odd-degree polynomials must have at least one real root.
Even-degree polynomials may have all complex roots (in conjugate pairs).

Factorization of real polynomials

Complete factorization theorem

Every polynomial with real coefficients can be factored into a product of:

Linear factors corresponding to real roots, and/or
Irreducible quadratic factors corresponding to complex conjugate pairs

In mathematical form, this means that a polynomial $P (x)$ may be factored as:

$P (x) = a_{n} (x - r_{1}) (x - r_{2}) \dots (x - r_{k}) \cdot (x^{2} + b_{1} x + c_{1}) (x^{2} + b_{2} x + c_{2}) \dots (x^{2} + b_{m} x + c_{m}),$

where:

$r_{1}, r_{2}, \dots, r_{k}$ are the real roots.
Each quadratic factor $x^{2} + b_{j} x + c_{j}$ is irreducible (has no real roots).
$k + 2 m = n$ (total degree).

Definitions

Irreducible: A quadratic $x^{2} + b x + c$ is irreducible over the reals if its discriminant is negative: $b^{2} - 4 c < 0$ .

The roots of such a quadratic are:

$x = \frac{- b \pm b ^{2} - 4 c}{2} = \frac{- b \pm i 4 c - b ^{2}}{2}$

These are complex conjugates, $α \pm β i$ , where $α = - \frac{b}{2}$ and $β = \frac{4 c - b ^{2}}{2}$ .

Next | 1.1.4 Logarithms

Polynomial functions

A polynomial function is a function of the form:

$P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0},$

where:

$a_{n}, a_{n - 1}, \dots, a_{1}, a_{0}$ are constants called coefficients,
$n$ is a non-negative integer called the degree of the polynomial, and
$a_{n} \neq = 0$ (the leading coefficient).

The fundamental theorem of algebra

The fundamental theorem of algebra is one of the most important results in mathematics:

Every polynomial of degree $n \geq 1$ with complex coefficients has exactly $n$ roots (counting multiplicities) in the complex numbers.

Its key implications include:

Existence of roots. Every non-constant polynomial has at least one root.
Complete factorization. Any polynomial can be completely factored into linear factors over the complex numbers.
Counting roots. A polynomial of degree $n$ has exactly $n$ roots when multiplicities are counted.

Polynomials with real coefficients

When a polynomial has real coefficients, which is the case in most applications in science and engineering, its roots have special properties:

Complex conjugate root theorem

If a polynomial with real coefficients has a complex root $a + bi$ (where $b \neq = 0$ ), then its complex conjugate $a - bi$ is also a root.

The consequences of this theorem include:

Complex roots come in pairs for polynomials with real coefficients.
Odd-degree polynomials must have at least one real root.
Even-degree polynomials may have all complex roots (in conjugate pairs).

Factorization of real polynomials

Complete factorization theorem

Every polynomial with real coefficients can be factored into a product of:

Linear factors corresponding to real roots, and/or
Irreducible quadratic factors corresponding to complex conjugate pairs

In mathematical form, this means that a polynomial $P (x)$ may be factored as:

$P (x) = a_{n} (x - r_{1}) (x - r_{2}) \dots (x - r_{k}) \cdot (x^{2} + b_{1} x + c_{1}) (x^{2} + b_{2} x + c_{2}) \dots (x^{2} + b_{m} x + c_{m}),$

where:

$r_{1}, r_{2}, \dots, r_{k}$ are the real roots.
Each quadratic factor $x^{2} + b_{j} x + c_{j}$ is irreducible (has no real roots).
$k + 2 m = n$ (total degree).

Definitions

Irreducible: A quadratic $x^{2} + b x + c$ is irreducible over the reals if its discriminant is negative: $b^{2} - 4 c < 0$ .

The roots of such a quadratic are:

$x = \frac{- b \pm b ^{2} - 4 c}{2} = \frac{- b \pm i 4 c - b ^{2}}{2}$

These are complex conjugates, $α \pm β i$ , where $α = - \frac{b}{2}$ and $β = \frac{4 c - b ^{2}}{2}$ .

FE Mechanical

1. Mathematics

1.1. Analytic geometry

Our FE Mechanical course is currently in development and is a work-in-progress.

Polynomial functions

3 min read

Font

Discuss

Feedback

A polynomial function is a function of the form:

$P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0},$

where:

$a_{n}, a_{n - 1}, \dots, a_{1}, a_{0}$ are constants called coefficients,
$n$ is a non-negative integer called the degree of the polynomial, and
$a_{n} \neq = 0$ (the leading coefficient).

The fundamental theorem of algebra

The fundamental theorem of algebra is one of the most important results in mathematics:

Every polynomial of degree $n \geq 1$ with complex coefficients has exactly $n$ roots (counting multiplicities) in the complex numbers.

Its key implications include:

Existence of roots. Every non-constant polynomial has at least one root.
Complete factorization. Any polynomial can be completely factored into linear factors over the complex numbers.
Counting roots. A polynomial of degree $n$ has exactly $n$ roots when multiplicities are counted.

Polynomials with real coefficients

When a polynomial has real coefficients, which is the case in most applications in science and engineering, its roots have special properties:

Complex conjugate root theorem

If a polynomial with real coefficients has a complex root $a + bi$ (where $b \neq = 0$ ), then its complex conjugate $a - bi$ is also a root.

The consequences of this theorem include:

Complex roots come in pairs for polynomials with real coefficients.
Odd-degree polynomials must have at least one real root.
Even-degree polynomials may have all complex roots (in conjugate pairs).

Factorization of real polynomials

Complete factorization theorem

Every polynomial with real coefficients can be factored into a product of:

Linear factors corresponding to real roots, and/or
Irreducible quadratic factors corresponding to complex conjugate pairs

In mathematical form, this means that a polynomial $P (x)$ may be factored as:

$P (x) = a_{n} (x - r_{1}) (x - r_{2}) \dots (x - r_{k}) \cdot (x^{2} + b_{1} x + c_{1}) (x^{2} + b_{2} x + c_{2}) \dots (x^{2} + b_{m} x + c_{m}),$

where:

$r_{1}, r_{2}, \dots, r_{k}$ are the real roots.
Each quadratic factor $x^{2} + b_{j} x + c_{j}$ is irreducible (has no real roots).
$k + 2 m = n$ (total degree).

Definitions

Irreducible: A quadratic $x^{2} + b x + c$ is irreducible over the reals if its discriminant is negative: $b^{2} - 4 c < 0$ .

The roots of such a quadratic are:

$x = \frac{- b \pm b ^{2} - 4 c}{2} = \frac{- b \pm i 4 c - b ^{2}}{2}$

These are complex conjugates, $α \pm β i$ , where $α = - \frac{b}{2}$ and $β = \frac{4 c - b ^{2}}{2}$ .

Next | 1.1.4 Logarithms

Polynomial functions

A polynomial function is a function of the form:

$P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0},$

where:

$a_{n}, a_{n - 1}, \dots, a_{1}, a_{0}$ are constants called coefficients,
$n$ is a non-negative integer called the degree of the polynomial, and
$a_{n} \neq = 0$ (the leading coefficient).

The fundamental theorem of algebra

The fundamental theorem of algebra is one of the most important results in mathematics:

Every polynomial of degree $n \geq 1$ with complex coefficients has exactly $n$ roots (counting multiplicities) in the complex numbers.

Its key implications include:

Existence of roots. Every non-constant polynomial has at least one root.
Complete factorization. Any polynomial can be completely factored into linear factors over the complex numbers.
Counting roots. A polynomial of degree $n$ has exactly $n$ roots when multiplicities are counted.

Polynomials with real coefficients

When a polynomial has real coefficients, which is the case in most applications in science and engineering, its roots have special properties:

Complex conjugate root theorem

If a polynomial with real coefficients has a complex root $a + bi$ (where $b \neq = 0$ ), then its complex conjugate $a - bi$ is also a root.

The consequences of this theorem include:

Complex roots come in pairs for polynomials with real coefficients.
Odd-degree polynomials must have at least one real root.
Even-degree polynomials may have all complex roots (in conjugate pairs).

Factorization of real polynomials

Complete factorization theorem

Every polynomial with real coefficients can be factored into a product of:

Linear factors corresponding to real roots, and/or
Irreducible quadratic factors corresponding to complex conjugate pairs

In mathematical form, this means that a polynomial $P (x)$ may be factored as:

$P (x) = a_{n} (x - r_{1}) (x - r_{2}) \dots (x - r_{k}) \cdot (x^{2} + b_{1} x + c_{1}) (x^{2} + b_{2} x + c_{2}) \dots (x^{2} + b_{m} x + c_{m}),$

where:

$r_{1}, r_{2}, \dots, r_{k}$ are the real roots.
Each quadratic factor $x^{2} + b_{j} x + c_{j}$ is irreducible (has no real roots).
$k + 2 m = n$ (total degree).

Definitions

Irreducible: A quadratic $x^{2} + b x + c$ is irreducible over the reals if its discriminant is negative: $b^{2} - 4 c < 0$ .

The roots of such a quadratic are:

$x = \frac{- b \pm b ^{2} - 4 c}{2} = \frac{- b \pm i 4 c - b ^{2}}{2}$

These are complex conjugates, $α \pm β i$ , where $α = - \frac{b}{2}$ and $β = \frac{4 c - b ^{2}}{2}$ .