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Chapter 2: Problem 77

What is a polynomial function?

Short Answer

Expert verified
A polynomial function is a mathematical function in which the variable only has positive integer powers. The terms of the function contain a variable and numerical coefficients. Example: \( P(x) = 2x^3 - 5x^2 + x + 7 \). The degree of the polynomial plays a key role - it's the highest power in the function and indicates the maximum number of solutions the function can have.

Step by step solution

01

- Identifying a Polynomial Function

A Polynomial function is a type of function that involves only positive integer powers of the variable. It is expressed as \( P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0 \), where \( x \) represents the variable, \( a_n, a_{n-1},..., a_1, a_0 \) are coefficients, with \( a_n 鈮� 0 \), and \( n \) is a nonnegative integer which represents the highest power of \( x \), also known as the degree of the polynomial.
02

- Understanding the terms of a polynomial

Every term of the polynomial function, \( a_nx^n, a_{n-1}x^{n-1},..., a_1x, a_0 \), is either a constant or a product of a constant coefficient and the variable raised to a whole number power. A polynomial with one term is called a monomial, with two terms - binomial, and with three terms - trinomial.
03

- The Degree of a Polynomial

The degree of the polynomial is the highest power of the variable. It dictates the most number of solutions or roots the polynomial equation can have. For instance, a polynomial of degree 3 may have up to 3 solutions.
04

- Examples of Polynomials

Consider these examples: 1. \( P(x) = 2x^3 - 5x^2 + x + 7 \): This is a polynomial of third degree. 2. \( P(x) = 4x + 3 \): This is a polynomial of first degree or linear polynomial. 3. \( P(x) = 5 \): This is a polynomial of zero degree or constant polynomial.

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Most popular questions from this chapter

Use the position function $$s(t)=-16 t^{2}+v_{0} t+s_{0}$$ \(\left(v_{0}=\text { initial velocity }, s_{0}=\text { initial position, } t=\text { time }\right)\) to answer Exercises. You throw a ball straight up from a rooftop 160 feet high with an initial velocity of 48 feet per second. During which time period will the ball's height exceed that of the rooftop?

You drive from your home to a vacation resort 600 miles away. You return on the same highway. The average velocity on the return trip is 10 miles per hour slower than the average velocity on the outgoing trip. Express the total time required to complete the round trip, \(T\), as a function of the average velocity on the outgoing trip, \(x .\)

Find the horizontal asymptote, if there is one, of the graph of rational function. $$g(x)=\frac{15 x^{2}}{3 x^{2}+1}$$

Write equations in point-slope form, slope-intercept form, and general form for the line passing through (-2,5) and perpendicular to the line whose equation is \(y=-\frac{1}{4} x+\frac{1}{3}\).

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+x-6}{x-3}$$
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