Problem 38 - Algebra and Trigonometry

Question

In Exercises 33–38, use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)=4 x^{4}-x^{3}+5 x^{2}-2 x-6 $$

Step-by-Step Solution

Answer

The given function can have either 3 or 1 positive real zeros and 1 negative real zero.

Step 1: Identify the coefficients and check the changes in sign for positive roots

Given function is \(f(x)=4 x^{4}-x^{3}+5 x^{2}-2 x-6\). The coefficients of the function are \(4, -1, 5, -2,-6\). Here, the sign of the coefficients changes 3 times (from positive to negative or negative to positive) (from \(4\) to \(-1\), then from \(-1\) to \(5\) and finally from \(5\) to \(-2\)). So, according to Descartes’s Rule of Signs, the possible number of positive real zeros can be 3 or 1 (it is less than the number of sign changes by an even number).

Step 2: Identify the coefficients and check the changes in sign for negative roots

To find the number of negative roots, replace \(x\) with \(-x\) in the function, and then see how many times the signs of coefficients change. When we replace \(x\) with \(-x\), we get \(f(-x) = 4(-x)^4 - (-x)^3 + 5(-x)^2 - 2(-x) - 6 = 4x^4 + x^3 + 5x^2 + 2x - 6\). In this case, the sign changes 1 time (from \(-6\) to \(2\)). Therefore, the possible number of negative real zeros is 1, as per Descartes’s Rule of Signs.

Step 3: Summarize the results

Combine all the results together. In this case, the given function can have either 3 or 1 positive real roots and 1 negative real root.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. These functions are of great importance in algebra. They can be simple expressions like a constant or linear term, or more complicated forms with multiple terms and variables.

In the given exercise, the polynomial function is:\[f(x) = 4x^4 - x^3 + 5x^2 - 2x - 6\]This particular function is a quartic polynomial because the highest power of the variable, which is the degree of the polynomial, is 4. Polynomial functions can be broken down into terms, each comprising a coefficient and a variable raised to an exponent. Understanding what these elements are and how they interact is crucial for solving polynomial-related problems.

Real Zeros

Real zeros of a polynomial function refer to the values of the variable for which the entire polynomial equals zero. They are also known as the roots of the equation. Finding these zeros is key to understanding the behavior and graph of the function.

• Positive real zeros are those that are greater than zero.
• Negative real zeros are less than zero.

In the context of the exercise, Descartes’s Rule of Signs is employed to determine how many positive and negative real zeros the polynomial might have. Identifying these zeros helps in graphing the polynomial function and analyzing its behavior across the number line.

Sign Changes

The concept of sign changes is central to Descartes's Rule of Signs. It involves observing how the signs of the coefficients in a polynomial change as one moves from one term to the next.

• A sign change occurs when consecutive coefficients differ in sign, indicating a transition from positive to negative or vice versa.
• The number of sign changes gives insight into the possible number of positive real zeros according to Descartes's Rule of Signs.

For the polynomial \(f(x) = 4x^4 - x^3 + 5x^2 - 2x - 6\), the sign changes occur between the terms: +4 to -1, -1 to +5, and +5 to -2. This results in three sign changes, suggesting up to three positive real zeros, following the rule that the number of real positive zeros is either equal to the number of variations or less than it by an even number.

Positive and Negative Roots

Positive and negative roots, or zeros, are identified by analyzing a polynomial's sign changes and modifying the variable's sign. Descartes's Rule of Signs is instrumental in determining the number of such roots.For positive roots:

• Count the sign changes in the original polynomial function's coefficients.

For negative roots:

• Replace the variable with its negative equivalent \((-x)\) in the polynomial and observe the new sign changes.

In the exercise, analyzing \(f(x)\) for positive roots yielded three sign changes indicating 3 or 1 positive zeros. For negative roots, substituting \(-x\) changed the polynomial to \(f(-x) = 4x^4 + x^3 + 5x^2 + 2x - 6\), resulting in one sign change and hence one negative zero. Understanding and applying these concepts allows you to find the possible distribution of zeros within a polynomial function geometrically and algebraically.