Textbook Question
Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x3−11x2+7x−5;f(4)
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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 32
Given f(x) = 2x^3 - 7x^2 + 9x - 3, use the Remainder Theorem to find f(- 13).
Verified step by step guidance1
Identify the polynomial function given: \( f(x) = 2x^3 - 7x^2 + 9x - 3 \).
According to the Remainder Theorem, the remainder of the division of \( f(x) \) by \( x - c \) is \( f(c) \).
In this problem, we need to find \( f(-13) \), so \( c = -13 \).
Substitute \( x = -13 \) into the polynomial: \( f(-13) = 2(-13)^3 - 7(-13)^2 + 9(-13) - 3 \).
Calculate each term separately and then sum them to find the remainder, which is \( f(-13) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Remainder Theorem
The Remainder Theorem states that for a polynomial function f(x), the remainder of the division of f(x) by (x - c) is equal to f(c). This theorem allows us to evaluate the polynomial at a specific point without performing long division, simplifying the process of finding function values.
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Polynomial Functions
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, f(x) = 2x^3 - 7x^2 + 9x - 3 is a cubic polynomial, which means it has a degree of 3 and can have up to three real roots. Understanding the structure of polynomial functions is essential for applying the Remainder Theorem.
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Function Evaluation
Function evaluation involves substituting a specific value into a function to determine its output. In this context, evaluating f(-13) means replacing x in the polynomial with -13 and calculating the resulting value. This process is fundamental in applying the Remainder Theorem to find the remainder when the polynomial is evaluated at a given point.
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Related Practice
Textbook Question
Divide using synthetic division. (2x5−3x4+x3−x2+2x−1)/(x+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(4−x)(x−6)≤0
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Textbook Question
In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=x2+6x+3
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Textbook Question
Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=2x−x2+3
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Textbook Question
Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero.
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