Question

Solve the polynomial inequality. $$(x+2)\left(x^{2}-4 x+5\right) \geq 0$$

   Solve the polynomial inequality.
$$(x+2)\left(x^{2}-4 x+5\right) \geq 0$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 3, Problem 17

Instant Answer

Solved by Expert Ramzi Deek

05/05/2020

Step 1

The critical points are the roots of the equation $(x+2)(x^{2}-4x+5) = 0$. Show more…

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Solve the polynomial inequality. $$(x+2)\left(x^{2}-4 x+5\right) \geq 0$$

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Key Concepts

Polynomial Inequalities

This concept involves finding the values for which a polynomial expression is greater than, less than, or equal to zero. It requires understanding the behavior of the polynomial across the real number line and determining the intervals where the expression meets the inequality criteria.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into a product of simpler polynomials. This technique is critical when solving inequalities because it allows us to identify the zeros of the expression, which are the points where the sign of the polynomial may change.

Critical Points (Zeros of the Factors)

Critical points, obtained by setting each factor of the polynomial equal to zero, divide the number line into intervals. These points are essential because they indicate where the polynomial can change sign, helping to determine the solution to the inequality.

Quadratic Discriminant

The discriminant of a quadratic expression provides information about the nature of its roots. A discriminant greater than, equal to, or less than zero tells us whether the quadratic has two real roots, one real repeated root, or no real roots respectively, which is useful in analyzing the sign of the quadratic factor in an inequality.

Sign Chart Analysis

Sign chart analysis is a method used to test the sign of each factor in different intervals determined by the critical points. By combining the individual signs of the factors, one can determine the sign of the entire polynomial on each interval, leading to the solution of the inequality.

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