Textbook Question
Identify which graphs are not those of polynomial functions.
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 13
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. 2x2+x<15
Verified step by step guidance1
Start by rewriting the inequality so that one side is zero: subtract 15 from both sides to get \(2x^{2} + x - 15 < 0\).
Next, factor the quadratic expression \(2x^{2} + x - 15\) if possible. Look for two numbers that multiply to \(2 \times (-15) = -30\) and add to \(1\) (the coefficient of \(x\)).
Once factored, write the inequality as a product of two binomials less than zero, for example, \((ax + b)(cx + d) < 0\).
Determine the critical points by setting each factor equal to zero and solving for \(x\). These points divide the number line into intervals.
Test a value from each interval in the original inequality to see if it satisfies the inequality. Use this to identify which intervals are part of the solution set, then express the solution in interval notation and graph it on the number line.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Inequalities
Polynomial inequalities involve expressions where a polynomial is compared to another value using inequality symbols like <, >, ≤, or ≥. Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.
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Linear Inequalities
Factoring and Finding Critical Points
To solve polynomial inequalities, first rewrite the inequality in standard form and factor the polynomial if possible. The roots or zeros of the polynomial, called critical points, divide the number line into intervals where the polynomial's sign can be tested to determine where the inequality holds.
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Interval Notation and Graphing Solution Sets
After determining the intervals where the inequality is true, express the solution set using interval notation, which concisely describes all values satisfying the inequality. Graphing on a real number line visually represents these intervals, showing open or closed endpoints depending on the inequality.
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Related Practice
Textbook Question
In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x3+13x2−11x−15)/(3x2−x−3)
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Textbook Question
Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2x2−8x+3
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Textbook Question
Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.
As _____
Textbook Question
Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as the cube of z and inversely as y.
Textbook Question
In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function.