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Chapter 2: Problem 21

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

Short Answer

Expert verified
The rational zeros of the function \(g(x)=x^{3}-4 x^{2}-x+4\) are \(x=1\), \(x=-1\), and \(x=4\).

Step by step solution

01

List all possible rational roots

By the Rational Root Theorem, the possible rational roots of \(g(x)=x^{3}-4 x^{2}-x+4\) are given by \(卤p/q\), where \(p\) is a factor of the constant term (in this case, 4) and \(q\) is a factor of the leading coefficient (in this case, 1). The factors of 4 are 卤1, 卤2, 卤4. The factors of 1 are 卤1. Therefore, the possible rational roots of \(g(x)\) are 卤1, 卤2, 卤4.
02

Test each possible root

The next step is to substitute each possible rational root into the function. If \(g(x)\) equals zero for a given root, then that root is a zero of the function. Testing this gives the following results: \(g(1)= 1-4-1+4=0\), \(g(-1)= -1-4+1+4=0\), \(g(2)= 8-16-2+4=-6\), \(g(-2)= -8-16+2+4=-18\), \(g(4)= 64-64-4+4=0\), \(g(-4)= -64-64+4+4=-120\).
03

Identify the rational zeros

From the results in step 2, the rational zeros of the function are \(x=1\), \(x=-1\), and \(x=4\), as these are the values that make \(g(x)=0\).

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Most popular questions from this chapter

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. $$f(x)=x^{4}+6 x^{2}-27$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=x^{2}+10 x+17$$

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Prove that the complex conjugate of the product of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the product of their complex conjugates.

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