Calculus I > Precalculus Review
Review Objective
Basic Concepts of Functions
Defining a Function
Intervals
Implied Domain
Even and Odd Functions
Composite functions
Inverse Functions
Polynomial Functions
Linear Functions
Quadratic Functions
Polynomial Function
Rational Functions
Rational Functions
Vertical and Horizontal Asymptote
Exponential and Logarithm
Rational and Irrational Exponents
Exponential Functions
Logarithmic Functions
Laws of Logarithms
Trigonometric functions
Radian and right triangles
Trigonometric functions
Trigonometric functions on a coordinate system
Sine and cosine function
Tangent function
Trigonometric additions and subtractions
Trigonometric formulas
Inverse sine function
Inverse cosine function
Inverse tangent function

Polynomial Function

Polynomials. We define the $ n$-th power $ x^n$ of $ x$ by $ {x^n = \underbrace{x\times x\times\cdots \times x}_{\mbox{$n$ times}}}$. A polynomial is of the form $ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$. Each $ a_k x^k$ of the polynomial is a term, and $ a_k$ of the term is called a coefficient. In particular, the coefficient $ a_n$ is called the leading coefficient of the polynomial if $ a_n \neq 0$.

Factoring polynomials. In many cases we find by the method of “trial and error” that a polynomial of interest can be expressed as a product of polynomials of the form $ (x - a)$. Occasionally you may be able to use the factoring formula

$\displaystyle x^2 + (a + b) x + ab = (x + a)(x + b). $

For example we can use it to factor

$\displaystyle x^2 + x - 6 = x^2 + (3 + (-2))x + (3)(-2) = (x + 3)(x - 2). $

Also frequently used are the following product formulas:

\begin{displaymath} \begin{array}{ll} (a \pm b)^2 = a^2 \pm 2ab + b^2 &\hspac... ...0.6in} (a \pm b)(a^2 \mp ab + b^2) = a^3 \pm b^3 \end{array} \end{displaymath}

Polynomial functions. A function $ f$ is called a polynomial function of degree $ n$ if

$\displaystyle f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

with leading coefficient  $ a_n \neq 0$.

Synthetic division. Suppose that $ f$ is a polynomial function, and that $ c$ is any real number. By using the synthetic division we can always obtain the quotient

$\displaystyle q(x) = a_n x^{n-1} + b_{1} x^{n-2} + \cdots + b_{n-2} x + b_{n-1}, $

and the remainder $ r$. Together the polynomial $ f(x)$ can be expressed as

$\displaystyle f(x) = (x - c) q(x) + r.$

Remainder Theorem. The result of synthetic division immediately implies $ f(c) = r$. In particular, $ f(c) = 0$ if and only if $ (x - c)$ is a factor of $ f(x)$.

Zeros of polynomials. The zeros of a polynomial $ f(x)$ are the solutions of the equation $ f(x) = 0$. In particular, the polynomial (6) of degree $ n$ has exactly $ n$ zeros

$\displaystyle \alpha_1,\alpha_2,\ldots,\alpha_n, $

including possible multiplicities (that is, it could be $ {\alpha_1 = \alpha_2}$, for example). Then it can be expressed as

$\displaystyle f(x) = a_n(x - \alpha_1)(x - \alpha_2)\cdots(x - \alpha_n) $

where $ a_n$ is the leading coefficient of the polynomial.

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