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Chapter 3: Problem 77

What is a polynomial function?

Short Answer

Expert verified
A polynomial function is a mathematical expression consisting of variables and coefficients, executed through addition, subtraction and multiplication. Each term of the polynomial (monomial) is of the form \( ax^n \), where \(a\) is a constant, \(x\) is the variable, and \(n\) is a non-negative integer. The highest power of \(x\) is called the degree of the polynomial.

Step by step solution

01

Defining a Polynomial

A polynomial is a mathematical expression consisting of variables and coefficients that are combined using addition, subtraction and multiplication. No operation of division by a variable is considered in a polynomial. Also, any exponent of the variable must be a whole, non-negative number.
02

Structure of Polynomial

It is represented in the form \(ax^n + bx^{n-1} + cx^{n-2} + ... + zx + y\) , where \(a, b, c, ... , z, y\) are constants and \(n\) is a non-negative integer. Each term of the polynomial must be of the form \( ax^n \) where \(a\) is a constant, \(x\) is the variable and \(n\) is a non-negative integer. Each term is called a monomial and the highest power of x in the polynomial is called the degree of the polynomial.
03

Examples of Polynomial

Examples of polynomials are: 1. Linear Polynomial: \( ax + b \), where \(a\) and \(b\) are constants and \(a 鈮� 0\).2. Quadratic Polynomial: \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants and \(a 鈮� 0\).3. Cubic Polynomial: \( ax^3 + bx^2 + cx + d \), where \(a\), \(b\), \(c\), and \(d\) are constants and \(a 鈮� 0\).

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Most popular questions from this chapter

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=-2 x^{3}(x-1)^{2}(x+5)$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{x-\frac{1}{x}}{x+\frac{1}{x}} $$

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=(x+3)(x+1)^{3}(x+4)$$

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=-3(x-1)^{2}\left(x^{2}-4\right)$$

Write the equation of a rational function$$ f(x)=\frac{p(x)}{q(x)} \text {having the indicated properties in which the degrees} $$ of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. f has a vertical asymptote given by x=1, a slant asymptote whose equation is y=x, y -intercept at 2, and x -intercepts at -1 and 2.
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