Problem 8
Question
In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=x^{\frac{1}{3}}-4 x^{2}+7$$
Step-by-Step Solution
Verified Answer
\(f(x)=x^{\frac{1}{3}}-4x^2+7\) is not a polynomial function.
1Step 1: Identification of function type
Looking at the function \(f(x)=x^{\frac{1}{3}}-4x^2+7\), we can see that the first term has a fractional exponent. According to the definition of a polynomial function, which only includes nonnegative integer powers of the variable, this term excludes the function from being a polynomial function.
2Step 2: Conclusion
Therefore, \(f(x)=x^{\frac{1}{3}}-4x^2+7\) is not a polynomial function. Identifying the degree becomes irrelevant as the function itself is not a polynomial function.
Key Concepts
Nonnegative Integer PowersFractional ExponentDegree of a Polynomial
Nonnegative Integer Powers
Understanding polynomial functions begins with a grasp on one of their basic components – nonnegative integer powers. These powers, which are exponents, must always be whole numbers that are zero or positive. For example, in the expressions \(x^0\), \(x^1\), \(x^2\), and so forth, the exponents are all nonnegative integers.
This requirement is essential because it ensures the output of the polynomial, for any real number input, is well-defined and predictable. Polynomial functions can have multiple terms, but each term must include the variable raised exclusively to a nonnegative integer power. A term like \(x^{-1}\) or \(x^{1/2}\) does not meet this criterion and therefore, any function containing such a term is not considered a polynomial function.
To identify if a given function is a polynomial, one must examine each term to ensure that the exponent on the variable is a nonnegative integer. If even one term does not adhere to this rule, the function is disqualified from being a polynomial. This foundational concept of requiring nonnegative integer powers is what separates polynomial functions from other types of functions.
This requirement is essential because it ensures the output of the polynomial, for any real number input, is well-defined and predictable. Polynomial functions can have multiple terms, but each term must include the variable raised exclusively to a nonnegative integer power. A term like \(x^{-1}\) or \(x^{1/2}\) does not meet this criterion and therefore, any function containing such a term is not considered a polynomial function.
To identify if a given function is a polynomial, one must examine each term to ensure that the exponent on the variable is a nonnegative integer. If even one term does not adhere to this rule, the function is disqualified from being a polynomial. This foundational concept of requiring nonnegative integer powers is what separates polynomial functions from other types of functions.
Fractional Exponent
In contrast to nonnegative integer powers, a fractional exponent indicates that the variable is involved in a root. For instance, \(x^{\frac{1}{2}}\) is another way of writing \(\sqrt{x}\), and \(x^{\frac{1}{3}}\) correlates with \(\sqrt[3]{x}\). Fractional exponents convey the concept of extracting roots, which are not confined to being integer values.
When a function includes a term with a fractional exponent, it signals that the function does not belong to the family of polynomial functions. This is because such terms imply operations that extend beyond the basic operations (addition, subtraction, multiplication, and nonnegative integer exponents) that define polynomials.
Understanding the presence of fractional exponents is a key skill in the classification of functions. In the function \(f(x)=x^{\frac{1}{3}}-4x^2+7\), the term \(x^{\frac{1}{3}}\) means it's necessary to consider the cube root of \(x\), which is not covered by the polynomial functions' definition. As fractional exponents fall outside the scope of polynomial functions, any term with such an exponent precludes the entire function from being classified as a polynomial.
When a function includes a term with a fractional exponent, it signals that the function does not belong to the family of polynomial functions. This is because such terms imply operations that extend beyond the basic operations (addition, subtraction, multiplication, and nonnegative integer exponents) that define polynomials.
Understanding the presence of fractional exponents is a key skill in the classification of functions. In the function \(f(x)=x^{\frac{1}{3}}-4x^2+7\), the term \(x^{\frac{1}{3}}\) means it's necessary to consider the cube root of \(x\), which is not covered by the polynomial functions' definition. As fractional exponents fall outside the scope of polynomial functions, any term with such an exponent precludes the entire function from being classified as a polynomial.
Degree of a Polynomial
The degree of a polynomial is a fundamental characteristic that helps describe its behavior. It is defined as the highest power to which the variable is raised among all the terms in the polynomial, after combining like terms. For example, in a polynomial like \(3x^4 + 2x^3 - x + 6\), the degree is 4 since the highest exponent in the terms is 4, which corresponds to the term \(3x^4\).
The degree informs us about the end behavior of the polynomial function – how the function behaves as \(x\) approaches positive or negative infinity. It also impacts the number of potential turning points and the possible number of real zeros (x-intercepts) the function can have. Remarkably, a non-zero constant polynomial, such as 5 or -3, is actually of degree 0 as it can be understood as \(x^0\), which reaffirms the degree as the highest power of \(x\) present.
In the context of our exercise, identifying the degree is only relevant for polynomial functions. Since the function \(f(x)=x^{\frac{1}{3}}-4x^2+7\) includes a term with a fractional exponent, it is not a polynomial, and thus talking about its degree in the polynomial sense does not apply. A function needs to conform strictly to the definition of polynomials, including having nonnegative integer powers across all terms, to have a degree that carries the implications typically associated with polynomial functions.
The degree informs us about the end behavior of the polynomial function – how the function behaves as \(x\) approaches positive or negative infinity. It also impacts the number of potential turning points and the possible number of real zeros (x-intercepts) the function can have. Remarkably, a non-zero constant polynomial, such as 5 or -3, is actually of degree 0 as it can be understood as \(x^0\), which reaffirms the degree as the highest power of \(x\) present.
In the context of our exercise, identifying the degree is only relevant for polynomial functions. Since the function \(f(x)=x^{\frac{1}{3}}-4x^2+7\) includes a term with a fractional exponent, it is not a polynomial, and thus talking about its degree in the polynomial sense does not apply. A function needs to conform strictly to the definition of polynomials, including having nonnegative integer powers across all terms, to have a degree that carries the implications typically associated with polynomial functions.
Other exercises in this chapter
Problem 8
Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{2
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Use the four-step procedure for solving variation problems given on page 445 to solve Exercises 1–10. C varies jointly as \(A\) and \(T . C=175\) when \(A=2100\
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In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=4 x^{5}-8 x^{4}-x+2 $$
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Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{2
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