## Polynomials

**Definition:**

A **polynomial function** of a real variable \(x\), is a function defined by the sum:

\[ P(x) = \sum_{k=0}^{n}c_k x^{k} = c_0 + c_1 x + c_2 x^{2}... + c_n x^{n}. \]

A linear function, \(f(x)\), of a real number, \(x\), is defined by two properties:

- for any real number, \(a\), \(f(ax) = a(f(x));\)
- for any real number, \(h\), \(f(x+h) = f(x) + f(h).\)

The product of two linear factors

\[ (x - \alpha)(x - \beta)\]

is a polynomial of degree two:

\[(x - \alpha)(x - \beta) = x^{2} - (\alpha + \beta)x + \alpha\beta.\]

**Definition:**

Polynomials of degree two are also called **quadratic polynomials**.

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**Definition:**

The **roots** of a polynomial \(P\) are the solutions of the equation:

\[ P(x) = 0. \]

How do we divide polynomials?

Suppose, for example that you are given the problems:

Express \(\frac{x^{2} - 1 } {x - 1 } \) in terms of \(x\).

Find \(\frac{x^{3} + 2x^{2} + x}{x + 1}\) in terms of \(x.\)

What do we do?