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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 8
Chapter 4, Problem 8

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x1/3 −4x2+7

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Recall that a polynomial function is a function of the form f(x) = anxn + an-1xn-1 + ... + a0 where each exponent n is a non-negative integer (0, 1, 2, ...).
Examine each term of the function f(x) = x1 3 − 4x2 + 7: the first term is x1 3, the second term is 4x2, and the third term is the constant 7.
Check the exponents of each term: the first term has an exponent of 1 3 (which is a fraction), the second term has an exponent of 2 (which is a non-negative integer), and the constant term can be considered as x0.
Since the first term has a fractional exponent, it violates the definition of a polynomial function, which requires all exponents to be whole numbers (non-negative integers).
Therefore, conclude that f(x) = x1 3 − 4x^2 + 7 is not a polynomial function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Functions

A polynomial function is a function that can be expressed as a sum of terms consisting of variables raised to non-negative integer exponents multiplied by coefficients. For example, f(x) = 3x^2 - 5x + 7 is a polynomial, but functions with variables under roots or with fractional exponents are not considered polynomials.
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Introduction to Polynomial Functions

Exponents and Their Restrictions in Polynomials

In polynomial functions, the exponents of the variable must be whole numbers (0, 1, 2, ...). Fractional or negative exponents indicate the function is not a polynomial. For instance, x^(1/3) is not allowed in polynomials because 1/3 is not an integer.
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Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. It determines the general shape and behavior of the polynomial graph. For example, in f(x) = -4x^2 + 7, the degree is 2 because the highest power of x is 2.
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Related Practice
Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.


As x→−3−x\(\to\)-3^{-}, f(x)→f\(\left\)(x\(\right\))\(\to\)____

Textbook Question

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. x2+5x+4>0

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Textbook Question

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. x2−6x+9<0

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Textbook Question

Divide using long division. State the quotient, and the remainder, r(x)r(x). 2x3+7x2+9x−20x+3\(\frac{2x^3 + 7x^2 + 9x - 20}{x + 3}\)

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Textbook Question

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as x and z. y = 25 when x = 2 and z = 5. Find y when x = 8 and z = 12.

Textbook Question

Find the domain of each rational function. f(x)=(x+7)/(x2+49)

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