# MandysNotes

## Polynomials

### The Definition of a Polynomial in One Real Variable

08 March 2014 By

Definition:

A  polynomial function of a real variable $$x$$, is a function defined by the sum:
$P(x) = \sum_{k=0}^{n}c_k x^{k} = c_0 + c_1 x + c_2 x^{2}... + c_n x^{n}.$

### Linear Functions and Linear Equations

08 March 2014 By

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, $$h$$, $$f(x+h) = f(x) + f(h).$$

### Polynomials of Degree Two

09 March 2014 By

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.

### The Roots of a Polynomial

09 March 2014 By

Definition:

The roots of a polynomial $$P$$ are the solutions of the equation:

$P(x) = 0.$

### Long Division with Polynomials

05 July 2014 By

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?