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Chapter 3: Problem 94

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, I only pay attention to the sign of \(f\) at each test value and not the actual function value.

Short Answer

Expert verified
The statement makes partial sense. While solving polynomial inequalities, indeed the sign of the function at each test value is crucial as it helps in determining the intervals where the function is positive or negative. However, it's not true that actual function values are completely ignored. They are evaluated when testing intervals. So, both the sign and function values cumulatively help in solving the inequality.

Step by step solution

01

Understand the statement

It is mentioned that to solve the inequality \(f(x)>0,\) where \(f\) is a polynomial function, only the sign of \(f\) at each test value is considered, not the actual function value.
02

Analyze the statement

Solving for polynomial inequalities involves factoring the polynomial, finding critical numbers (roots or zeros of the polynomial), determining intervals, and testing the intervals. So, when determining the solution for an inequality such as \(f(x) > 0\), we actually look for regions on the number line where the function is positive. This requires understanding not only the value of function at particular points but also how it behaves around those points.
03

Conclude whether the statement makes sense or not

The statement partially makes sense as during the process of solving polynomial inequalities, the sign of \(f\) at each test value is indeed important. However, it is not completely true that the actual function value is ignored. As while testing intervals, the function is evaluated at specific points from each interval which directly involves function values. But the end goal indeed is to determine where the function is positive (or negative as required).

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