Textbook Question
Solve each polynomial inequality. Give the solution set in interval notation. x4 + 6x2 + 1 ≥ 4x3 + 4x
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Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 46
If the given term is the dominating term of a polynomial function, what can we conclude about each of the following features of the graph of the function? (a) domain (b) range (c) end behavior (d) number of zeros (e) number of turning points -9x6
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Identify the dominating term of the polynomial function, which is given as \(-9x^{6}\). This term determines the overall behavior of the polynomial for very large or very small values of \(x\).
For the domain (a), recall that polynomial functions are defined for all real numbers, so the domain is \((-\infty, \infty)\) regardless of the dominating term.
For the range (b), analyze the leading term \(-9x^{6}\). Since the exponent 6 is even and the coefficient is negative, the function will tend to negative infinity as \(|x|\) becomes large, indicating the range will be bounded above but extend downwards without bound.
For the end behavior (c), because the leading term is \(-9x^{6}\), as \(x \to \infty\) or \(x \to -\infty\), the function \(f(x) \to -\infty\). This means both ends of the graph fall downward.
For the number of zeros (d) and turning points (e), remember that a polynomial of degree 6 can have up to 6 real zeros and up to 5 turning points. The dominating term's degree gives these maximum possible counts, but the actual number depends on the full polynomial.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Dominating Term of a Polynomial
The dominating term of a polynomial is the term with the highest degree, which determines the overall shape and behavior of the graph for large values of x. In this case, -9x^6 is the dominating term, meaning the polynomial behaves like -9x^6 as x approaches infinity or negative infinity.
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Introduction to Polynomials
Domain and Range of Polynomial Functions
The domain of any polynomial function is all real numbers since polynomials are defined everywhere. The range depends on the leading term's degree and sign; for an even degree with a negative leading coefficient like -9x^6, the function tends to negative infinity as x grows large, so the range is limited above.
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End Behavior, Zeros, and Turning Points
End behavior describes how the function behaves as x approaches ±∞, dictated by the leading term. For -9x^6, both ends go to negative infinity. The number of zeros is at most the degree (6), and the number of turning points is at most one less than the degree (5), reflecting where the graph changes direction.
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Related Practice
Textbook Question
Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = x2 +2x -8; k=2
Textbook Question
Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(4x2+25)/(x2+9)
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Textbook Question
Several graphs of the quadratic function ƒ(x) = ax2 + bx + c are shown below. For the given restrictions on a, b, and c, select the corresponding graph from choices A–F. (Hint: Use the discriminant.) a < 0; b2 - 4ac < 0
Textbook Question
Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=x4+3x3-3x2-11x-6
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Textbook Question
For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = 6x4 + x3 - 8x2 + 5x+6; k=1/2