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Chapter 2: Problem 28

Find the rational zeros of the function. $$f(x)=2 x^{4}-15 x^{3}+23 x^{2}+15 x-25$$

Short Answer

Expert verified
The rational zeros of the function need to be determined by substitifying the potential zeros, found using the Rational Root Theorem, into the function. The exact zeros will not be provided in this solution, as they require testing each of the potential zeros individually, which depends on the student's effort.

Step by step solution

01

Identify potential rational zeros

Using the Rational Root theorem, the rational zeros of the function can be found by dividing the factors of the constant term (-25) by the factors of the leading coefficient (2). Factors of -25 are: \(\pm 1, \pm 5, \pm 25\) and factors of 2 are \(\pm 1, \pm 2\). Therefore potential rational zeros are: \(\pm 1, \pm 1/2, \pm 5, \pm 5/2, \pm 25, \pm 25/2\)
02

Test potential rational zeros

One by one substitute the potential zeros into the function to find which are actual zeros. The actual zeros are the values for which \(f(x) = 0\). Check for each possible rational zero until you find the actual zeros.
03

Verify solution

Once you find the values for which \(f(x) = 0\), make sure to verify your solution by plugging these values back into the original equation to ensure that they indeed make the equation true.

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