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Chapter 3: Problem 95

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm solving a polynomial inequality that has a value for which the polynomial function is undefined.

Short Answer

Expert verified
The statement does not make sense because polynomial functions, and by extension polynomial inequalities, are defined for all real numbers. A situation where a polynomial function would be undefined at a certain value is not a typical characteristic of polynomials.

Step by step solution

01

Understanding Polynomials

Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are typically expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_2, a_1, a_0\) are constants. The crucial point here is that polynomials are defined for all real numbers because you can perform these operations for any real number value of \(x\).
02

Analyzing the Statement

The statement declared that there is a value for which the polynomial function is undefined. Under normal circumstances, this situation does not occur for polynomial functions because polynomials are always defined for any real number. If a polynomial function was undefined at a point, it would mean that point creates an undefined operation like division by zero, but polynomial don't have division operation. Hence, the statement does not make sense.
03

Application to Polynomial Inequalities

A polynomial inequality is a relation of the form \(P(x) > 0\) or \(P(x) < 0\), where \(P(x)\) is a polynomial. The analysis of the statement also applies to polynomial inequalities. The concept that a polynomial (and by extension, a polynomial inequality) would be undefined at a certain point does not hold since polynomials are defined for every real number.

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