Textbook Question
Factor ƒ(x) into linear factors given that k is a zero.
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 21
Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=5x5+2x3-3x+4
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Identify the leading term of the polynomial function. For the given function \(f(x) = 5x^5 + 2x^3 - 3x + 4\), the leading term is \$5x^5\( because it has the highest power of \)x$.
Determine the degree and the leading coefficient of the polynomial. Here, the degree is 5 (an odd number), and the leading coefficient is 5 (a positive number).
Recall the general end behavior rules for polynomials based on degree and leading coefficient:
- If the degree is odd and the leading coefficient is positive, then as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).
- If the degree is odd and the leading coefficient is negative, the end behaviors are reversed.
- If the degree is even and the leading coefficient is positive, both ends go to \(\infty\).
- If the degree is even and the leading coefficient is negative, both ends go to \(-\infty\).
Apply the rule to the given polynomial: since the degree is odd (5) and the leading coefficient is positive (5), the graph falls to negative infinity on the left and rises to positive infinity on the right.
Use the end behavior diagram notation to describe this: as \(x \to -\infty\), \(f(x) \to -\infty\) (downward arrow), and as \(x \to \infty\), \(f(x) \to \infty\) (upward arrow). This can be represented as \(\downarrow \quad \uparrow\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Degree and Leading Term
The degree of a polynomial is the highest power of the variable in the expression, and the leading term is the term with this highest power. The degree and leading coefficient determine the general shape and end behavior of the polynomial's graph.
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Standard Form of Polynomials
End Behavior of Polynomial Functions
End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. It depends primarily on the degree (even or odd) and the sign of the leading coefficient, indicating whether the graph rises or falls at the ends.
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Using End Behavior Diagrams
End behavior diagrams visually represent the direction of the graph's ends using arrows or symbols. They help summarize the polynomial's behavior at extreme values of x, making it easier to understand and communicate the function's long-term trends.
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Related Practice
Textbook Question
Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=-x3-4x2+2x-1
Textbook Question
Use synthetic division to perform each division. (x3 - 1) / (x-1)
Textbook Question
If ƒ(x) is a polynomial function with real coefficients, and if 7+2i is a zero of the function, then what other complex number must also be a zero?
Textbook Question
Use synthetic division to perform each division. x4-1 / x-1
Textbook Question
Graph the following on the same coordinate system.
(a) y = (x - 2)2
(b) y = (x + 1)2
(c) y = (x + 3)2
(d) How do these graphs differ from the graph of y = x2?