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Chapter 3: Problem 81

What is a polynomial inequality?

Short Answer

Expert verified
A polynomial inequality is a type of mathematical inequality that involves a polynomial. It is typically solved by first rearranging it into a standard form, then determining the critical values by solving the corresponding equation, and finally testing points in each interval determined by the critical values. The solution set is the range of the variable values that make the inequality true.

Step by step solution

01

Definition

A polynomial inequality is an inequality that involves a polynomial. Inequalities are mathematical expressions involving the symbols > (greater than), < (less than), 鈮� (greater than or equal to), and 鈮� (less than or equal to). When these symbols are used with polynomials, a polynomial inequality is formed.
02

Examples of Polynomial Inequalities

Here are some examples of polynomial inequalities: \[ x^2 - 3x + 2 > 0 \], \[ 3x^3 - 2x^2 + x - 1 鈮� 0 \], \[ -x^4 + 7x^2 + 2 < 0 \]. Typically, the solution to a polynomial inequality is a range of values for the variable that makes the inequality true.
03

Solving Polynomial Inequalities

To solve a polynomial inequality, you would first need to rearrange the inequality so that it reads 'expression > 0', 'expression < 0', 'expression 鈮� 0', or 'expression 鈮� 0', where 'expression' is a polynomial. The next step is then to identify the critical values of the polynomial, which are obtained by setting the polynomial equal to zero and solving for the variable. Then, by testing points in each interval determined by the critical values in the inequality, it would be possible to find the solution set.

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Most popular questions from this chapter

Determine whether cach statement is true or false If bhe statement is false, make the necessary change(s) to produce a true statement. It is possible to have a rational function whose graph has no \(y\) -intercept.

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies directly as \(x . y=45\) when \(x=5 .\) Find \(y\) when \(x=13 .\)

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies jointly as \(m\) and the square of \(n\) and inversely as \(p\) \(y=15\) when \(m=2, n=1,\) and \(p=6 .\) Find \(y\) when \(m=3, n=4,\) and \(p=10\).

The perimeter of a rectangle is 180 feet. Describe the possible lengths of a side if the area of the rectangle is not to exceed 800 square feet.

Whe lise a graphing utility to graph $$ f(x)=\frac{x^{2}-4 x+3}{x-2} \text { and } g(x)=\frac{x^{2}-5 x+6}{x-2} $$ What differences do you observe between the graph of \(f\) and the graph of \(g\) ? How do you account for these differences?
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