Powers and Polynomials

Powers

For \(a \in \mathbb{R}\) and \(n \in \mathbb{N}\), the power \(a^n\) is defined as:

\[ a^n = \underbrace{a \cdot a \cdot \ldots \cdot a}_{n \text{ factors}}, \quad a^0 = 1 \;(a \neq 0) \]

Here \(a\) is the base and \(n\) is the exponent.

Laws of Exponents

For \(a, b \neq 0\) and \(m, n \in \mathbb{Z}\):

Law	Formula
Same base, multiplication	\(a^m \cdot a^n = a^{m+n}\)
Same base, division	\(\frac{a^m}{a^n} = a^{m-n}\)
Power of a power	\((a^m)^n = a^{m \cdot n}\)
Product	\((a \cdot b)^n = a^n \cdot b^n\)
Quotient	\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)
Negative exponent	\(a^{-n} = \frac{1}{a^n}\)

Example. \(2^3 \cdot 2^4 = 2^7 = 128\).

Example. \((3^2)^3 = 3^6 = 729\).

Polynomials

A polynomial in the variable \(x\) is an expression of the form:

\[ p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 = \sum_{k=0}^{n} a_k x^k \]

The numbers \(a_0, a_1, \ldots, a_n\) are called coefficients. The coefficient \(a_n\) (\(a_n \neq 0\)) is the leading coefficient.

Degree

The degree of a polynomial is the highest exponent with a non-zero coefficient:

\[ \deg(p) = n \quad \text{if } a_n \neq 0 \]

Examples.

• \(p(x) = 3x^4 - 2x + 7\) has degree \(4\).
• \(p(x) = 5\) (constant, \(\neq 0\)) has degree \(0\).
• The zero polynomial \(p(x) = 0\) has no defined degree (convention: \(\deg(0) = -\infty\)).

Rules for the Degree

\[ \deg(p \cdot q) = \deg(p) + \deg(q) \]

\[ \deg(p + q) \leq \max(\deg(p), \deg(q)) \]

Roots

A root (or zero) of \(p(x)\) is a value \(x_0\) with \(p(x_0) = 0\). A polynomial of degree \(n\) has at most \(n\) roots (over \(\mathbb{R}\); counted with multiplicity over \(\mathbb{C}\), exactly \(n\)).

Example. \(p(x) = x^2 - 4 = (x-2)(x+2)\) has roots \(x = 2\) and \(x = -2\).

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Summary

Concept	Definition
\(a^n\)	\(a\) multiplied by itself \(n\) times
Polynomial	\(\sum_{k=0}^n a_k x^k\)
Degree	Highest exponent with \(a_k \neq 0\)
Root	\(x_0\) with \(p(x_0) = 0\)

References

• Lang, Serge: Algebra. Springer, 3rd edition, 2002. Chapter 4.