Problem 50 - 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. $$ 3 x^{4}-11 x^{3}-3 x^{2}-6 x+8=0 $$

Step-by-Step Solution

Answer

The zeros of the polynomial equation \(3 x^{4}-11 x^{3}-3 x^{2}-6 x+8 = 0\) are \( x = 1\), \(x = -\frac{2}{3}\) and \(x = 2\).

Step 1: Finding Potential Rational Zeros

The Rational Zero Theorem states that, if the polynomial has a rational zer0 \(p/q\), then \(p\) is a factor of the constant term (in this case, 8), and \(q\) is a factor of the leading coefficient (in this case, 3). Therefore, all possible rational zeros for this polynomial are \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 1/3, \pm 2/3, \pm 4/3, \pm 8/3\).

Step 2: Descartes Rule of Signs

The number of sign changes in the polynomial gives the possible number of positive real zeros. In this case, there are two sign changes in \(3 x^{4}-11 x^{3}-3 x^{2}-6 x+8\), suggesting 2 or 0 positive real zeros. When it comes to the possible number of negative real zeros, substituted \(x\) with \(-x\) into the polynomial equation, encapsulating \(3(-x)^{4}-11(-x)^{3}-3(-x)^{2}-6(-x)+8\), and obtaining \(3 x^{4}+11 x^{3}-3 x^{2}-6 x+8\). There are 3 sign changes, suggesting 3 or 1 negative real zeros.

Step 3: Testing Potential Zeros

Start with smaller values from the list of potential zeros identified in Step 1 and accurately substitute into the polynomial. If the polynomial equals 0, then that value is a zero. Repeat this for all values in the list. Here, we can find the following zeros: \(x = 1\), \(x = -\frac{2}{3}\), and \(x = 2\).

Step 4: Confirming Zeros Using Synthetic Division

By replacing the zeros found in step 3 using synthetic division with the polynomial \(3 x^{4}-11 x^{3}-3 x^{2}-6 x+8\), we can confirm that those are indeed zeros of the polynomial as a result.

Key Concepts

Rational Zero Theorem

Understanding the Rational Zero Theorem is crucial for anyone dealing with polynomial equations. It's quite a straightforward yet powerful tool that provides a list of potential rational zeros of a polynomial function.

The theorem asserts that if a polynomial has rational zeros, then each zero can be expressed as a fraction \( \frac{p}{q} \), where \( p \) is a factor of the trailing constant term and \( q \) is a factor of the leading coefficient. For example, if our polynomial's constant term is 8 and the leading coefficient is 3, the possible values of \( p \) and \( q \) are the factors of 8 (\(\pm 1, \pm 2, \pm 4, \pm 8\)) and 3 (\(\pm 1, \pm 3\)), respectively.

This means the potential rational zeros might be \( \pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}, \pm 8, \pm \frac{8}{3} \). This list is a starting point for identifying the actual zeros. It's essential to test these potential zeros to confirm which ones are indeed zeros of the polynomial.

Descartes's Rule of Signs

Descartes's Rule of Signs is yet another valuable method for predicting the number of positive and negative real zeros in a polynomial function without actually solving the equation.

The technique involves counting the number of times the coefficients of the terms change signs. For instance, in the polynomial \( 3x^4 - 11x^3 - 3x^2 - 6x + 8 \) there are 2 sign changes (from positive to negative, and then from negative to positive), indicating there could be 2 or 0 positive real zeros. This is because the Rule of Signs states that the actual number of positive real zeros is either equal to the number of sign changes or less than that by an even number.

Similarly, by substituting \( x \) with \( -x \) and analyzing the sign changes of the resulting polynomial, one can deduce the number of negative real zeros. This rule narrows down the possibilities significantly, saving time and effort in finding the actual solutions.

Synthetic Division

Synthetic division is a simplified method of dividing polynomials, particularly useful when dividing by a linear factor. It is less cumbersome than traditional long division and provides a quick way to check if a number is a zero of a polynomial function.

Essentially, synthetic division involves using the coefficients of the polynomial and the potential zero to perform a series of operations that will confirm whether the value is indeed a zero. If the final value obtained after performing synthetic division is zero, it affirms that the tested value is a root of the polynomial equation.

In our example, once potential zeros are identified, synthetic division is used to verify them as actual zeros of the polynomial \( 3x^4 - 11x^3 - 3x^2 - 6x + 8 \). This process helps not only to confirm the zeros but also to reduce the polynomial's degree, simplifying the problem further.

Polynomial Function

At its core, a polynomial function is a mathematical expression made up of terms consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation.

A polynomial is generally expressed in the form \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where each \( a_i \) represents a coefficient, \( x \) is a variable, and \( n \) indicates the degree of the polynomial defined by the highest power of \( x \) with a non-zero coefficient.

The fundamental aspect of polynomial functions that makes them so interesting and useful is the relationship between their algebraic properties and their graphical representations. Understanding the nature of polynomial zeros and how they relate to the function's graph is crucial in various fields, from engineering to economics.