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Chapter 13: Problem 5

Find all the roots of \(3 x^{3}+x^{2}-11 x+6=0\) with an accuracy of two decimal places.

Short Answer

Expert verified
The roots of the equation are approximately \(x \approx 1\), \(x \approx 1.79\), and \(x \approx -1.12\) rounded to two decimal places.

Step by step solution

01

Identify Potential Rational Roots

First, by the Rational Root Theorem, the potential rational roots of the polynomial are of the form \(p/q\), where \(p\) is a factor of the constant term (6) and \(q\) is a factor of the leading coefficient (3). This gives us the possible rational roots: \(\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{3}, \pm\frac{2}{3}\).
02

Test Potential Rational Roots

Use synthetic division or direct substitution to test each potential rational root. Find one that is indeed a root, for instance, testing \(x=1\), the equation simplifies to \(3 + 1 - 11 + 6 = 0\) so \(x=1\) is a root.
03

Factor Out the Root

After finding a valid root, factor out the corresponding factor \(x - 1\) from the polynomial. Synthetic division or long division can be used here, and you should get a quadratic equation as the quotient.
04

Verify the Roots

Check that each candidate for the roots, including those from the quadratic formula solution, satisfies the original equation to the required accuracy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Root Theorem
The Rational Root Theorem is a powerful tool for finding the possible rational roots of a polynomial equation where the coefficients are integers. When you have a polynomial equation like \(3 x^{3}+x^{2}-11 x+6=0\), the theorem says that any rational root can be written as \(p/q\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.

In our exercise, the constant term is 6, and the leading coefficient is 3. The possible factors of 6 (\

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