Problem 63 - College Algebra

Question

Which one of the following is true? a. The equation \(x^{3}+5 x^{2}+6 x+1=0\) has one positive real root. b. Descartes's Rule of Signs gives the exact number of positive and negative real roots for a polynomial equation. c. Every polynomial equation of degree 3 has at least one rational root. d. None of the above is true.

Step-by-Step Solution

Answer

Option d 'None of the above is true' is the correct choice.

Step 1: Consider option a

First, examine the first statement: 'The equation \(x^{3}+5 x^{2}+6 x+1=0\) has one positive real root.' Using Descartes's Rule of Signs, change of signs in this polynomial from + to - and - to + is counted. Here, there are no such changes, indicating 0 positive roots. Hence, this statement is false.

Step 2: Consider option b

Now, consider the second statement: 'Descartes's Rule of Signs gives the exact number of positive and negative real roots for a polynomial equation.' The keyword here is 'exact'. Descartes's Rule of Signs gives the possible number of positive and negative real roots, but not always the exact number. Therefore, this statement is also untrue.

Step 3: Consider option c

The third statement is: 'Every polynomial equation of degree 3 has at least one rational root'. This is a misunderstanding of the Rational Root Theorem. The theorem suggests that if a polynomial has a rational root, then this root can be expressed as a ratio of factors of the constant term and factors of the leading coefficient. However, not every cubic equation has a rational root. Therefore, this statement is false as well.

Step 4: Consider option d

Given a, b, and c are incorrect, the only remaining option is d, 'None of the above is true,' which is indeed correct, as the other claims are incorrect.

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions that set a polynomial equal to a value, typically zero. These equations form the basis of countless mathematical problems and applications across various fields.

For instance, the polynomial equation in the exercise, \(x^{3}+5x^{2}+6x+1=0\), sets a cubic polynomial equal to zero and is looking for the values of \(x\) that satisfy this condition, referred to as its 'roots'. Polynomial equations can be linear, quadratic, cubic, or of a higher degree, with their complexity and the strategies to solve them increasing with degree.

It's crucial to understand that the highest exponent in the equation gives the highest number of possible real roots. In this example, a cubic equation should have three roots in total, which could be real or complex numbers. Here, tools like Descartes's Rule of Signs are used to estimate the number of positive and negative real roots.

Rational Root Theorem

The Rational Root Theorem is a handy tool for solving polynomial equations, especially when targeting rational roots. It states that if a given polynomial \(f(x)\) has integer coefficients and a rational root \(\frac{p}{q}\), where \(p\) and \(q\) have no common factors, then \(p\) must be a factor of the constant term and \(q\) must be a factor of the leading coefficient.

This theorem provides a systematic way to list all possible rational roots that can be tested as actual roots of the polynomial. However, it does not guarantee the existence of rational roots, but rather gives a set of candidates that could be if they exist at all.

For the polynomial \(x^{3}+5x^{2}+6x+1=0\), for example, one might misinterpret the Rational Root Theorem to say it must have a rational root because it's of degree 3; however, as seen in the exercise, this is not the case. The theorem merely suggests potential rational roots, not their assured existence.

Real Roots of Polynomials

Discovering the real roots of polynomials can often be a challenging endeavor. The real roots are the solutions to the polynomial equation that are not imaginary numbers. Various techniques, including factoring, graphing, and using algorithms like Descartes's Rule of Signs, can help estimate or identify these roots.

Descartes's Rule of Signs specifically is a powerful heuristic that provides the possible number of positive and negative real roots of a polynomial by counting the number of sign changes in the sequence of its coefficients. However, it is vital to recognize that this rule only gives the maximum number of positive or negative roots and does not confirm their exact amount, which is a common misconception.

In the context of our exercise, the rule indicated zero sign changes for the given polynomial, suggesting there are no positive real roots, which further helped us disprove one of the proposed statements.