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Chapter 2: Problem 3

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$f(x)=3 x^{4}-11 x^{3}-x^{2}+19 x+6$$

Short Answer

Expert verified
The possible rational zeros for the given function \(f(x)=3 x^{4}-11 x^{3}-x^{2}+19 x+6\) as per the Rational Zero Theorem are 卤1, 卤2, 卤3, 卤6, 卤1/3, 卤2/3.

Step by step solution

01

Identify the coefficients

The polynomial given here is \(f(x)=3 x^{4}-11 x^{3}-x^{2}+19 x+6\). Here, \(a_{n} = 3\) and \(a_{0} = 6\).
02

List all factors

Now list all positive and negative factors of \(a_{n}\) and \(a_{0}\). The factors of 3 are: 卤1, 卤3. The factors of 6 are: 卤1, 卤2, 卤3, 卤6.
03

Find all possible rational zeros

Now form all the possible fractions by dividing each factor of \(a_{0}\) by each factor of \(a_{n}\). The possible rational zeros are 卤1/1, 卤2/1, 卤3/1, 卤6/1, 卤1/3, 卤2/3, 卤1, 卤2, 卤3, 卤6, 卤1/3, 卤2/3, 卤3/3, 卤6/3.
04

Eliminate duplicates

After eliminating the duplicates, the final list of possible rational zeros of the given function is 卤1, 卤2, 卤3, 卤6, 卤1/3, 卤2/3.

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