Problem 45 - Algebra and Trigonometry

Question

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes’s Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ x^{4}-3 x^{3}-20 x^{2}-24 x-8=0 $$

Step-by-Step Solution

Answer

The zeros of the given polynomial function are \(x = -2, 2, -2, 4\)

Step 1: Apply the Rational Zero Theorem

Using the Rational Zero theorem, which states that any rational zero of a polynomial equation can be written in the form \(\pm p/q\), where p is a factor of the constant term and q is a factor of the leading coefficient. Here, the constant term is -8 and the leading coefficient is 1. The possible rational zeros of the polynomial can be \(\pm 1, \pm2, \pm 4, \pm 8\).

Step 2: Find the number of positive and negative real roots using Descartes Rule of Signs

By looking at the number of sign changes in the polynomial function, we can find the number of positive real roots. Here, there is 3 sign changes (from positive to negative and from negative to positive). So, there are 3 or 1 positive real roots. If we replace x with -x, we will have \( -x^{4}-3 x^{3}-20 x^{2}- 24x- 8 =0 \), then there are 2 sign changes in the polynomial function, suggesting there are 2 or 0 negative real roots.

Step 3: Test all possible rational zeros

One way to test for which of these rational roots are actual roots is to substitute each into the polynomial and see if it yields 0. However, a more effective method is using synthetic division to conduct these tests. After conducting the synthetic division with all possible rational zeros, we see that when we tried applying -2, the remainder was zero. Hence, -2 is a zero of the polynomial. Given this root, the function factors to \(-2\) and a cubic function \(x^{3}-x^{2}-16x-4\).

Step 4: Solve the remaining cubic function

To find the remaining roots, we must solve for the cubic function \(x^{3}-x^{2}-16x-4 = 0\) using the above methodologies or other appropiate resolution techniques for cubic equations. In this case, we find the remaining roots are \(2, -2, 4\).

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem is a crucial tool for solving polynomial equations. It predicts that the possible rational zeros (fractions) of a polynomial equation with integer coefficients are of the form \( \pm p/q \), where \( p \) is a factor of the constant term (the term without the variable \( x \) ) and \( q \) is a factor of the leading coefficient (the coefficient of the highest power of \( x \)).

For example, consider the polynomial \( x^{4} - 3x^{3} - 20x^{2} - 24x - 8 \). The constant term is -8, and the leading coefficient is 1. Applying the Rational Zero Theorem gives us a list of possible rational zeros: \( \pm1, \pm2, \pm4, \pm8 \). While not all these values will be actual zeros, this theorem narrows down the possibilities considerably, facilitating the solution process.

Descartes's Rule of Signs

Descartes's Rule of Signs is another valuable principle for determining the number of positive and negative real roots that a polynomial might have. By analyzing the number of times the coefficients change signs, we can predict the maximum number of positive real roots. Additionally, by substituting \( -x \) for \( x \) and analyzing the sign changes again, we can find out the possible number of negative real roots.

Considering the same polynomial as before, when the signs change three times, this indicates there can be 3 or 1 positive real roots. After substituting \( -x \) for \( x \), and getting two sign changes, it suggests there can be 2 or 0 negative real roots. This rule gives us a guideline but does not guarantee the exact number of real roots.

Synthetic Division

Synthetic division is a simplified method of performing polynomial long division when dividing by a linear factor. It helps to determine whether a given number is a zero of the polynomial. By systematically testing the possible rational zeros derived from the Rational Zero Theorem, we can find actual zeros with less effort.

For instance, when applying synthetic division to our example polynomial with \( -2 \) which is one of the possible rational zeros, if the remainder is zero, then \( -2 \) is indeed a root. Finding one zero then allows us to factor the polynomial and reduce it to a simpler equation, facilitating the discovery of the remaining roots.

Cubic Equations

Cubic equations are polynomial equations of degree three. Solving these equations can be more complex and often requires different strategies such as factoring by grouping, the Rational Zero Theorem, synthetic division, or other advanced techniques like Cardano's formula.

After finding the first zero and reducing our example polynomial, we are left with a cubic equation. With the Rational Zero Theorem and synthetic division, we can continue testing for zeros. Once we find all the rational zeros, we can then factor the cubic equation completely to find its solutions. In this case, the solutions or zeros of the reduced cubic equation are \( 2, -2, \text{and} 4 \).