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. (x−4)(x+2)>0
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 1
Determine which functions are polynomial functions. For those that are, identify the degree.
Verified step by step guidance1
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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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 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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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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05:16Standard 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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Related Practice
Textbook Question
In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x) = (x + 4)^2 - 2
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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. (x+3)(x−5)>0
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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 directly as x. y = 65 when x = 5. Find y when x = 12.
Textbook Question
Divide using long division. State the quotient, and the remainder, r(x). (x2+8x+15)÷(x+5)
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Textbook Question
In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
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