Textbook Question
The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
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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 5
Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=4x4−x3+5x2−2x−6
Verified step by step guidance1
Identify the polynomial function: \(f(x) = 4x^{4} - x^{3} + 5x^{2} - 2x - 6\).
Recall the Rational Zero Theorem: any rational zero, expressed as \(\frac{p}{q}\), must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient.
List the factors of the constant term \(-6\): \(\pm1, \pm2, \pm3, \pm6\).
List the factors of the leading coefficient \(4\): \(\pm1, \pm2, \pm4\).
Form all possible rational zeros by taking each factor of the constant term over each factor of the leading coefficient, simplifying if possible, and include both positive and negative values.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Zero Theorem
The Rational Zero Theorem helps identify all possible rational zeros of a polynomial function by considering factors of the constant term and the leading coefficient. Possible rational zeros are formed by ±(factors of constant term) divided by (factors of leading coefficient). This theorem narrows down candidates for roots before testing them.
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Polynomial Functions and Degree
A polynomial function is an expression involving variables raised to whole-number exponents with coefficients. The degree of the polynomial is the highest exponent, which determines the maximum number of zeros the function can have. Understanding the degree helps anticipate the number of possible roots.
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Factors of Integers
Factors of an integer are numbers that divide it exactly without leaving a remainder. To apply the Rational Zero Theorem, you must find all positive and negative factors of the constant term and leading coefficient. This step is essential to list all possible rational zeros systematically.
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Related Practice
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 and inversely as the square of z. y = 20 when x = 50 and z = 5. Find y when x = 3 and z = 6.
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Textbook Question
Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x5−x4−7x3+7x2−12x−12
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Textbook Question
Divide using long division. State the quotient, and the remainder, .
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Textbook Question
In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x4−11x3−3x2−6x+8
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Textbook Question
Find the domain of each rational function. h(x)=(x+7)/(x2−49)
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