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Chapter 2: Problem 71

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=x^{3}-3 x^{2}+x+5$$

Short Answer

Expert verified
The polynomial can't be factored using integer coefficients and no rational roots were found using the Rational Root Theorem. One would have to use the cubic formula or a calculator to find the exact roots and subsequently the linear factors of the polynomial.

Step by step solution

01

Set the Polynomial Equal to Zero

First step is to set the polynomial equal to zero to find its roots: \(x^{3}-3x^{2}+x+5 = 0\)
02

Rational Root Theorem

We can use the Rational Root Theorem to determine potential rational roots of our polynomial. The Rational Root Theorem states that any potential rational zero, p/q, of a polynomial must have p as a factor of the constant term and q as a factor of the coefficient of the leading term. In this case, potential rational roots could be factors of 5, which are \(\pm 1\), and \(\pm 5\).
03

Test Potential Roots

The next step is to substitute each potential root into the polynomial equation to find the true roots. In this case we find that none of the potential roots are in fact actual roots to the equation.
04

Using Cubic Formula

For a third degree polynomial with no rational roots, we could use the cubic formula to find the roots. However, the computation is quite complex and generally requires a scientific calculator.
05

Compile the Results

At the end when we find the roots, our linear factors will be \(x-\) each root and we list out all the roots found.

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Most popular questions from this chapter

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{1}{4} x^{2}-x+\frac{1}{4}=\frac{1}{4}\left(4 x^{3}-x^{2}-4 x+1\right)$$

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.

Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) $$5,3-2 i$$

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x+4=0$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{2}+36$$
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