Polynomials

A polynomial is a mathematical expression made up of variables, coefficients, and exponents that are non-negative integers, combined using addition, subtraction, and multiplication.

Example: 4x^3 - 2x^2 - 7x - 5 A Polynomial is defined through it's coefficients. compact form: p(x) = \sum_{i=0}^{n} a_i x^i

Monom

A monom is a type of polynomial that has exactly one term.

Example: ax^{n}

Binomial

A binomial is a type of polynomial that has exactly two terms.

Example: 3a^2 - 7

Trinomial

A trinomial is a type of polynomial that has exactly three terms.

Example: 2a^2 - 3a + 8

Constant term (Glied)

The term without any x, i.e., the standalone number in the polynomial. If f: \mathbb{R} \to \mathbb{R}, x \mapsto 4x^4 - 8x^2 +19 the constant term would be 19.

Leading coefficient (Leitkoeffizient)

This refers to coefficient at the beginning of the polynomial. If we look at f: \mathbb{R} \to \mathbb{R}, x \mapsto 4x^4 - 8x^2 +19 again the leading coefficient would be 4.

degree(p) (grad(p)): Highest exponent of a term. The coefficient of the term has to be greater then 0 - Example: p(x) = 0 x^4 + 3 x^3 + 1 x^2 + 7 x + 10; degree(p) = 3 - Example: p(x) = 21; degree(p)=0 Equivalent: Two Polynomials are equivalent if they have the same degree (d) and \forall i \in {0,...d} : a_{i} = b_{i}

Zeropolynomial: No coefficient. - Example: p(x) = 0; degree(p) = -\infty

Addition

g(x) = p(x) + f(x) = c_n x^n + \ldots + c_0 with c_i = (a_i + b_i). All missing coefficient are getting replaced with 0.

The degree of a polynomial does not change through addition: g(x) := p(x) + f(x); degree(g) \leq max{degree(p),degree(f)}. Polynomial addition is commutative.

Example: \begin{align} p(x) + f(x) &= (4+13) x^4 + 7 x^3 + 2 x^2 + (7+1) x + (4+43)\\ &= 17 x^4 + 7 x^3 + 2 x^2 + 8 x + 47 \end{align}

Subtraction

g(x) = p(x) + f(x) = c_n x^n + \ldots + c_0 with c_i = (a_i - b_i). All missing coefficient are getting replaced with 0.

The degree of a polynomial can only degrees through subtraction.

Example: \begin{align} p(x) - f(x) &= (4-4) x^4 + (0-7) x^3 -5 x^2 + (1-1) x + (11-43)\\ &= -7 x^3 -5 x^2 - 32 \end{align}

Multiplikation

When multipliying with a monom:

\begin{align} p(x) \cdot q(x) &= (a_n x^n + \ldots + a_0) (b x^m)\\ &= (a_n \cdot b) x^{n+m} + \ldots + (a_1 \cdot b) x^{1+m} + (a_0 \cdot b) x^{m} \end{align}

Example: \begin{align} p(x) \cdot q(x) &= (4 x^4 + 5 x^2 + x + 11) (7 x^3)\\ &= (4 \cdot 7) x^{4+3} + (5 \cdot 7) x^{2+3} + (1 \cdot 7) x^{1+3} + (11 \cdot 7) x^{0+3}\\ &= 28 x^7 + 35 x^5 + 7 x^4 + 77 x^3 \end{align}

otherwise:

\begin{align} d&:=degree(p) \\\\ p(x) \cdot f(x) &= \sum_{i=0}^d p_i \cdot f(x) \end{align}

If g(x) = p(x) \cdot f(x)

1. degree(g) \leq degree(p) + degree(f)
2. degree(g) \geq degree(p)
3. degree(g) \geq degree(f)

Example: \begin{align} p(x) \cdot f(x) &= 4 x^4 (2 x^3 + 3 x) + 7x^2 (2 x^3 + 3 x) + 4 ( 2 x^3 + 3 x)\\ &= 8 x^7 + 12 x^5 + 14 x^5 + 21 x^3 + 8 x^3 + 12 x\\ &= 8 x^7 + 26 x^5 + 29 x^3 + 12 x\\ \end{align}

Division

1. Sort both in polynomials by their power and write them down (p(x) \div q(x) = \dots).
2. Divide (\div)
3. Multiply (\cdot)
4. Subtract (-)
5. If degree(r) < degree(q) go back to step 2

\begin{align} \frac{p(x)}{q(x)} = s(x) + \frac{r(x)}{q(x)} \\\\ degree(r) < degree(q) \end{align}