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Chapter 2: Problem 77

What is a polynomial function?

Short Answer

Expert verified
A polynomial function is a mathematical function of a variable (or variables) where the variable only appears with non-negative integer exponents and the function is a sum of these variable terms. Represented in general form, it is \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0 \).

Step by step solution

01

Understanding Polynomials

A polynomial function is a type of mathematical function where the variables are only raised to non-negative integer values. The function can be described as a sum of terms, where each term is a variable with a coefficient.
02

Structure of a Polynomial Function

A polynomial function, generally, can be represented as \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0 \), where \( n \) is a non-negative integer and \( a_0, a_1, ..., a_n \) are coefficients.
03

Characteristics of Polynomial Functions

Some characteristics of polynomial functions are that they are continuous and smooth, meaning that they have no sharp corners or discontinuities. A polynomial of degree \( n \) has at most \( n \) real roots and \( n-1 \) turning points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degree of a Polynomial
When dealing with polynomial functions, understanding the degree is crucial. The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial function \( f(x) = 4x^3 + 3x^2 - 2x + 7 \), the highest power is 3, so the degree is 3.

  • The degree indicates the polynomial's behavior as \( x \) becomes very large or very small.
  • A higher degree usually implies a more complex curve with more turns.
  • The degree offers insights into the number of roots and shape of the polynomial's graph.
Therefore, the degree is not only a piece of numeric data; it has substantial implications on the function's visual and numeric characteristics.
Coefficients
Coefficients are the numerical factors that multiply the variable terms within a polynomial. In the polynomial \( g(x) = 5x^4 - 2x^3 + 3x - 1 \), the coefficients are 5, -2, 3, and -1.

  • Each coefficient affects the polynomial's graph by altering its slope or curve at various points.
  • Changing a coefficient changes the amplitude of that part of the polynomial.
  • The coefficient of the highest degree term, known as the leading coefficient, heavily influences the end behavior of the polynomial.
Playing around with coefficients can drastically alter the shape and roots of a polynomial, allowing us to model different types of situations or data.
Real Roots
In polynomial functions, real roots are the \( x \) values where the polynomial equals zero; these are the solutions to the equation \( f(x) = 0 \). For example, in \( h(x) = x^2 - 4x + 4 \), the root is \( x = 2 \) because \( (2)^2 - 4(2) + 4 = 0 \).

  • A polynomial of degree \( n \) can have at most \( n \) real roots.
  • Real roots correspond to the x-intercepts on the graph of the polynomial.
  • Multiple roots mean the graph touches or crosses the x-axis multiple times at the same point.
Identifying the real roots is critical in graphing the polynomial and solving real-world problems it may model.
Continuity
Polynomials are continuous functions, meaning there are no breaks, holes, or sharp corners in their graphs. This property ensures that the curve of the polynomial can be drawn without lifting the pencil from the paper.

  • Continuity implies that as \( x \) approaches any value, the function approaches a specific value \( f(x) \).
  • This is vital in calculus for analyzing limits and derivatives.
  • Continuous functions can accurately model real-life situations where abrupt changes are unnatural or undesirable.
Being continuous makes polynomial functions ideal for mathematical modeling across various disciplines.
Smoothness
The term smoothness in polynomial functions refers to the absence of sharp corners or cusps. Polynomials are inherently smooth due to their continuous nature and the fact that their derivatives of all orders are also continuous.

  • Unlike polynomials, a sharp corner in a graph would suggest a point where the function is not differentiable.
  • Smooth curves ensure predictable behavior and easy mathematical handling.
  • Models involving smooth functions are often more realistic for natural phenomena.
Smoothness guarantees gradual, logical changes in the function and is a favorable property when analyzing and predicting behaviors in physical systems.

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Most popular questions from this chapter

Use long division to rewrite the equation for \(g\) in the form quotient \(+\frac{\text { remainder }}{\text { divisor }}\) Then use this form of the function's equation and transformations of \(f(x)=\frac{1}{x}\) to graph \(g\) $$g(x)=\frac{2 x+7}{x+3}$$

Does the equation \(3 x+y^{2}=10\) define \(y\) as a function of \(x ?\) (Section \(1.2,\) Example 3 )

Use a graphing utility to graph $$ f(x)=\frac{x^{2}-4 x+3}{x-2} \text { and } g(x)=\frac{x^{2}-5 x+6}{x-2} $$ What differences do you observe between the graph of \(f\) and the graph of \(g\) ? How do you account for these differences?

An athlete whose event is the shot put releases the shot wilh the same initial velocity but at different angles. The figure shows the parabolic paths for shots released at angles of \(35^{\circ}\) and \(65^{\circ} .\) Exercises \(57-58\) are based on the functions that model the parabolic paths. (table cannot copy) Among all pairs of numbers whose difference is \(24,\) find a pair whose product is as small as possible. What is the minimum product?

If you have difficulty obtaining the functions to be maximized in Exercises \(73-76,\) read Example 2 in Section \(1.10 .\) On a certain route, an airline carries 8000 passengers per month, each paying \(\$ 50 .\) A market survey indicates that for each \(\$ 1\) increase in the ticket price, the airline will lose 100 passengers. Find the ticket price that will maximize the airline's monthly revenue for the route. What is the maximum monthly revenue?
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