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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 16
Chapter 4, Problem 16

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. 3x2+16x<−53x^2+16x<−5

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First, rewrite the inequality so that one side is zero. Add 5 to both sides to get: \$3x^{2} + 16x + 5 < 0$.
Next, factor the quadratic expression \$3x^{2} + 16x + 5$. To do this, look for two numbers that multiply to \(3 \times 5 = 15\) and add to 16.
Once you find the factors, express the quadratic as a product of two binomials: \((ax + b)(cx + d) < 0\).
Find 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 determine where the inequality holds true. Then, express the solution set in interval notation and graph it on the real 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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Factoring and Solving Quadratic Equations

To solve polynomial inequalities, especially quadratics, it is essential to rewrite the inequality in standard form and factor the polynomial if possible. Factoring helps find the roots (zeros) of the polynomial, which divide the number line into intervals to test for the inequality.
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Solving Quadratic Equations by Factoring

Interval Notation and Graphing Solution Sets

After determining the intervals where the inequality holds, solutions are expressed using interval notation, which concisely represents all values satisfying the inequality. Graphing on a number line visually shows these solution intervals, using open or closed circles to indicate whether endpoints are included.
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Related Practice
Textbook Question

In Exercises 15–18, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] <IMAGE>

f(x)=−x4+x2f(x)=−x^4+x^2

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Textbook Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−x2−2x+8

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Textbook Question

Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z and inversely as the square of w.

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Textbook Question

Find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=−2(x−1)(x+2)2(x+5)3f(x)=-2(x-1)(x+2)^2(x+5)^3

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Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As x→1+, f(x)→x\(\to\)1^{+},\(\text{ }\)f(x)\(\to\) __

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Textbook Question

Divide using long division. State the quotient, and the remainder, r(x).2x5−8x4+2x3+x22x3+1\(\frac{2x^5 - 8x^4 + 2x^3 + x^2}{2x^3 + 1}\)

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