Question

Solve each polynomial inequality. $$ x^{4}+4 \geq 5 x^{2} $$

   Solve each polynomial inequality.
$$
x^{4}+4 \geq 5 x^{2}
$$

Precalculus

Raymond A. Barnett,… 7th Edition

Chapter 4, Problem 51

Instant Answer

Solved by Expert James Kiss

07/14/2021

Step 1

We subtract $5x^2$ from both sides to get: $$ x^{4} - 5x^{2} + 4 \geq 0 $$ Show more…

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Solve each polynomial inequality. $$ x^{4}+4 \geq 5 x^{2} $$

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Key Concepts

Interval Testing

After determining the critical points, interval testing (or the sign chart method) is used to determine the sign of the polynomial on each interval. By testing values from each interval, one can ascertain where the polynomial satisfies the inequality conditions, thereby defining the solution set.

Critical Points

Critical points are the values of the variable at which the polynomial equals zero. These points partition the number line into intervals where the sign of the polynomial remains consistent. Identifying these roots is crucial because they serve as potential boundary points for the solution set of the inequality.

Polynomial Inequality

A polynomial inequality involves a polynomial expression that is compared to zero (or another value) using inequality signs. Solving such an inequality typically means finding the intervals of the variable for which the inequality holds by determining where the polynomial is positive, negative, or zero.

Rewriting the Inequality in Standard Form

Rewriting the inequality to have zero on one side is an essential initial step. This transforms the inequality into a standard form where one can analyze the sign of the entire polynomial expression, which is instrumental for further analysis.

Factoring Techniques

Factoring is a method used to simplify polynomial expressions into a product of simpler polynomials or factors. It often involves substitution methods for higher-degree polynomials, recognition of special products like perfect squares, and other algebraic identities. This simplification is key to identifying the roots or critical points of the inequality.

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