Solve the inequality. (x^3+3 x^2-9 x-27)/(x+4) ≤0

Name: Problem 25, Chapter 3: College Algebra
Uploaded: 2020-07-28T04:37:26-08:00
Duration: 3 min 41 s
Channel: Ankit Gupta
Description: Solve the inequality. (x^3+3 x^2-9 x-27)/(x+4) ≤0

Key Concepts

Rational Expressions

A rational expression is a fraction in which both the numerator and the denominator are polynomials. Understanding these expressions involves recognizing that the expression is undefined for any value that makes the denominator zero, and that such restrictions can lead to discontinuities in the function's graph.

Polynomial Factorization

Polynomial factorization is a technique used to break down a polynomial into simpler multiplied factors, which can help in identifying the zeros or roots of the polynomial. This process is crucial in solving equations and inequalities as it simplifies the expression and reveals critical points where the expression may change sign.

Domain Restrictions

Domain restrictions refer to the values that a variable cannot take, particularly values that cause the denominator of a rational expression to become zero. Recognizing these restrictions is vital when solving inequalities since they determine the intervals that need to be excluded from the solution set due to undefined behavior.

Critical Points

Critical points are the values of the variable that make either the numerator or the denominator of a rational expression equal to zero. These points divide the number line into intervals, on which the sign of the expression can be tested. They serve as boundaries in determining where the inequality holds true.

Sign Chart Analysis

Sign chart analysis is a method used to determine the sign of a rational expression in various intervals defined by its critical points. By selecting test points in each interval, one can deduce whether the expression is positive or negative in that region, which directly informs the solution of the inequality.

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