Textbook Question
Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. 5+i and 5-i
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Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 61
Solve each rational inequality. Give the solution set in interval notation.
Verified step by step guidance1
First, identify the rational inequality: \(\frac{x^2 - 3x - 4}{x^2 + 6x + 9} \leq 0\).
Factor both the numerator and the denominator: numerator \(x^2 - 3x - 4\) factors as \((x - 4)(x + 1)\), and denominator \(x^2 + 6x + 9\) factors as \((x + 3)^2\).
Determine the critical points by setting numerator and denominator equal to zero: numerator zeros at \(x = 4\) and \(x = -1\), denominator zero at \(x = -3\) (note this makes the denominator zero, so \(x = -3\) is excluded from the domain).
Create a number line and mark the critical points \(-3\), \(-1\), and \(4\). Test values from each interval formed by these points in the inequality to determine where the expression is less than or equal to zero.
Write the solution set by including intervals where the inequality holds true, remembering to exclude points where the denominator is zero, and express the solution in interval notation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Inequalities
Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the expression is positive, negative, or zero, considering the domain restrictions from the denominator.
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3:21
Nonlinear Inequalities
Factoring Polynomials
Factoring is the process of breaking down polynomials into products of simpler polynomials. It helps identify zeros of the numerator and denominator, which are critical points for determining intervals to test in the inequality.
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07:30Introduction to Factoring Polynomials
Interval Testing and Domain Restrictions
After finding critical points, the number line is divided into intervals. Each interval is tested to see if the inequality holds. Additionally, values that make the denominator zero are excluded from the solution set because they are undefined.
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3:51Domain Restrictions of Composed Functions
Related Practice
Textbook Question
Solve each rational inequality. Give the solution set in interval notation. (2x - 3)/(x + 1) > 4
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Textbook Question
Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=3x4+2x3-4x2+x-1; no real zero greater than 1
Textbook Question
Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.
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Textbook Question
Graph each rational function. ƒ(x)=(x+1)/(x-4)
Textbook Question
Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.
; no real zero less than -1