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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 1
Capitolo 4, Problema 1

Determine which functions are polynomial functions. For those that are, identify the degree. f(x)=5x2+6x3f(x)=5x^2+6x^3

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Recall that a polynomial function is a function of the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where each exponent is a non-negative integer and the coefficients \(a_i\) are real numbers.
Look at the given function: \(f(x) = 5x^2 + 6x^3\). Check the exponents of \(x\) in each term. Here, the exponents are 2 and 3, both of which are non-negative integers.
Since all terms have non-negative integer exponents and real coefficients, \(f(x)\) is a polynomial function.
To find the degree of the polynomial, identify the term with the highest exponent. In this case, the highest exponent is 3 from the term \$6x^3$.
Therefore, the degree of the polynomial function \(f(x) = 5x^2 + 6x^3\) is 3.

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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 powers multiplied by coefficients. For example, f(x) = 5x^2 + 6x^3 is a polynomial because the exponents are whole numbers and coefficients are real numbers.
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06:04
Introduction to Polynomial Functions

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the function with a non-zero coefficient. In f(x) = 5x^2 + 6x^3, the degree is 3 because the term 6x^3 has the highest exponent.
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Standard Form of Polynomials

Identifying Polynomial Terms

To determine if a function is polynomial, check each term's exponent to ensure it is a non-negative integer and that the function does not include variables in denominators, roots, or negative exponents. Terms like x^2 and x^3 qualify, confirming the function is polynomial.
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Identifying Intervals of Unknown Behavior
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