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Chapter 3: Problem 3

Determine which functions are polynomial functions. For those that are, identify the degree. \(g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x\)

Short Answer

Expert verified
The given function \(g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x\) is a polynomial function and its degree is \(5\).

Step by step solution

01

Identifying a polynomial function

A function \(g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x\) can be represented in the form \(f(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + ... + a_{1} x + a_{0}\), where each \(a\) is a real number and \(n\) is non-negative integers, therefore, \(g(x)\) is a polynomial function.
02

Identifying the degree of the polynomial

The degree of the polynomial is determined by the highest exponent of \(x\) in the polynomial equation. In this case, \(5\) is the highest exponent of \(x\), therefore the degree of the polynomial is \(5\).

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Key Concepts

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

Degree of Polynomial
The degree of a polynomial is a key concept in understanding polynomial functions. It refers to the highest power of the variable present in the polynomial expression. For example, consider a polynomial in the form:
  • \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\)
In this expression, the degree of the polynomial is \(n\), provided that \(a_n\) is non-zero.

Knowing the degree of a polynomial helps us understand its fundamental characteristics and behavior:
  • It indicates the maximum number of times the graph of the polynomial can intersect the x-axis, also known as its roots or zeros.
  • The degree suggests the potential number of turning points of the graph, which is at most \(n-1\).
  • It provides insight into the polynomial's end behavior or its behavior as \(x\) approaches infinity or negative infinity.
So, when looking at a polynomial like \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5} x\), the degree is \(5\) as the highest exponent is \(x^5\). This suggests a graph with potentially 5 roots and up to 4 turning points.
Identifying Polynomial Functions
Identifying whether a function is a polynomial function is an important first step in polynomial analysis. A polynomial function must adhere to certain criteria and take the standard form:
  • \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\)
Here, \(a_n, a_{n-1}, ..., a_0\) represent real numbers and \(n\) must be a non-negative integer. A function qualifies as a polynomial if:
  • All exponents of the variable are non-negative integers.
  • All coefficients of the exponents are real numbers.
  • It does not include variables in denominators, variables inside radicals, or variables raised to a negative or fractional power.
For \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5}x\), each term features a non-negative integer as an exponent (\(5, 3,\) and \(1\), respectively). Furthermore, all coefficients are real numbers, qualifying \(g(x)\) as a polynomial function.
Polynomial Degree Determination
Determining the degree of a polynomial is a straightforward process. You simply identify the term with the highest exponent of the variable \(x\). This number represents the polynomial's degree. Let's break down the process:
  • Step 1: List all terms of the polynomial and note the exponents of \(x\) in each term.
  • Step 2: Identify the highest exponent present in these terms.
  • Step 3: The term with the highest exponent dictates the degree of the polynomial.
In our example function \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5}x\), the exponents are \(5, 3,\) and \(1\). The greatest of these is \(5\), thus the polynomial's degree is \(5\). Recognizing the degree is crucial as it influences how we interpret the function's graph and behavior in various mathematical contexts.

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Most popular questions from this chapter

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