Polynomial Basics & Vector Spaces

Polynomials are the building blocks of computational mathematics. Before we can approximate, interpolate, or differentiate using polynomials, we need a solid understanding of their structure — what they are made of, how they relate to each other, and what space they inhabit.

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The Standard Polynomial

A polynomial of degree $n$ is written as:

$P_n(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n$

or equivalently using sigma notation:

$P_n(x) = \sum_{k=0}^{n} a_k x^k$

Property	Value
Degree	$n$, the highest power of $x$
Number of coefficients	$n + 1$

Examples

• $P_3(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3$ has degree 3 and 4 coefficients: $a_0, a_1, a_2, a_3$.
• A polynomial of degree 27 has $27 + 1 = \mathbf{28}$ coefficients.

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Polynomials as a Vector Space

Polynomials are not just formulas. They form a vector space. This means we can combine them in two fundamental ways:

Adding polynomials (vector addition):

$(1 + x + x^2) + x^3 = 1 + x + x^2 + x^3$

Multiplying by a scalar:

$(1 + x + x^2) \times 5 = 5 + 5x + 5x^2$

Both operations always produce another polynomial. This closure property is the defining feature of a vector space.

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Basis and Dimension

Basis

The basis of a polynomial space is the minimal set of polynomials that can generate every polynomial in the space through linear combination.

For the space $P_3(x)$, the standard basis is:

$\{1,\; x,\; x^2,\; x^3\}$

Any cubic polynomial, regardless of its coefficients, can be written as a linear combination of these four basis elements. With them, we can generate any polynomial of degree 3 or less.

Dimensional Space

The dimension of the space is the number of elements in its basis. For polynomials of degree $n$:

$\dim(P_n) = n + 1$

Polynomial	Degree	Dimension	Number of Coefficients
$P_3(x)$	3	4	4
$P_{27}(x)$	27	28	28
$P_{58}(x)$	58	59	59

Note

Dimension, degree, and coefficient count are all linked. A polynomial of degree $n$ lives in an $(n+1)$-dimensional space and always has exactly $n+1$ coefficients.

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Functional Space vs. Polynomial Space

A general function $f(x)$ can contain infinitely many terms:

$f(x) = 2 + 3x + 10x^2 + 14x^3 + 25x^4 + \cdots$

There is no upper bound on the degree, so $f(x)$ belongs to a vector space of infinite dimension:

$f(x) \in V^\infty$

A polynomial $P_n(x)$, by contrast, terminates at degree $n$. It lives in a finite-dimensional subspace:

$P_n(x) \in V^{n+1}$

When we perform polynomial approximation or interpolation, we are essentially projecting an infinite-dimensional function $f(x)$ into a finite-dimensional polynomial space $V^{n+1}$. The gap between $f(x)$ and $P_n(x)$ is the approximation error, and controlling that error is the central challenge of this chapter.

Caution

Because $P_n(x) \in V^{n+1}$ is a strict subspace of $V^\infty$, no finite polynomial can exactly represent a non-polynomial function. The best we can do is control how small the difference $|f(x) - P_n(x)|$ becomes, and the next section shows precisely how.