Problem 69 - Algebra and Trigonometry

Question

Use Descartes’ Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$ P(x)=x^{5}+4 x^{3}-x^{2}+6 x $$

Step-by-Step Solution

Answer

The polynomial can have 2 or 0 positive real zeros, and 0 negative real zeros. Total possible real zeros: 2 or 0.

Step 1: Write the Polynomial in Standard Form

Rewrite the polynomial in standard form: \( P(x) = x^5 + 0x^4 + 4x^3 - x^2 + 6x + 0 \). This will help identify the changes in sign needed for Descartes' Rule of Signs.

Step 2: Identify Sign Changes for Positive Zeros

Look at the coefficients of the polynomial \( P(x) \): \( 1, 0, 4, -1, 6 \). The changes in sign are from \( 0 \) to \( 4 \) (no sign change), from \( 4 \) to \( -1 \) (one change), and from \( -1 \) to \( 6 \) (another change). Therefore, there are 2 sign changes.

Step 3: Use Descartes’ Rule of Signs for Positive Zeros

According to Descartes' Rule of Signs, the number of positive real zeros is either equal to the number of sign changes or less than it by an even number. Hence, there can be 2 or 0 positive real zeros.

Step 4: Change Variable for Negative Zeros

To find sign changes for negative zeros, substitute \( x \) with \( -x \) in the polynomial, yielding \( P(-x) = (-x)^5 + 4(-x)^3 - (-x)^2 + 6(-x) = -x^5 - 4x^3 - x^2 - 6x \).

Step 5: Identify Sign Changes for Negative Zeros

For \( P(-x) = -x^5 - 4x^3 - x^2 - 6x \), the coefficients are \( -1, -4, -1, -6 \). Since all signs are negative, there are no sign changes.

Step 6: Use Descartes’ Rule of Signs for Negative Zeros

There are 0 sign changes, indicating there are 0 negative real zeros.

Step 7: Determine Possible Total Number of Real Zeros

According to the polynomial degree (5), and with 2 or 0 positive zeros and 0 negative zeros, the possible total number of real zeros is 2 or 0. The rest must be complex zeros if not real zeros.

Key Concepts

Polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables, multiplied by coefficients. In this exercise, our polynomial is expressed as:

• \( P(x) = x^5 + 4x^3 - x^2 + 6x \)

This expression is referred to as a polynomial because it consists of multiple terms that are combined using addition, subtraction, and multiplication.
The degree of a polynomial is determined by the highest power of the variable present. Here, the degree is 5, as the term with the highest exponent is \( x^5 \).
Understanding the structure and forming of a polynomial is crucial because it offers insight into how many zeros—solutions to the equation \( P(x) = 0 \)—the polynomial might have, as well as their nature.

Real Zeros

Real zeros of a polynomial are the values of \( x \) that make the polynomial equal to zero. They are essentially the x-coordinates where the graph of the polynomial touches or crosses the x-axis.

• If a polynomial has a real zero, \( P(x) \) evaluates to zero for that particular \( x \).
• The number of real zeros is affected by the polynomial's behavior and degree.

In our exercise, we are trying to find out how many such "x-values" exist, both positive and negative. Descartes' Rule of Signs helps to predict these values by analyzing sign changes in the polynomial's coefficients. It is a useful tool for determining the potential number of real zeros before solving the polynomial firsthand.

Sign Changes

The concept of sign changes involves observing where the polynomial's coefficients change from positive to negative or vice versa during their sequence. Each sign change has implications on the possible number of zeros.

• A sign change occurs when subsequent coefficients differ in sign.
• Observing these changes provides clues about potential roots.

To find sign changes:- First express the polynomial in standard form, checking each consecutive coefficient for sign differences. - In our exercise, sign changes were identified by comparing coefficients in \( P(x) = x^5 + 0x^4 + 4x^3 - x^2 + 6x \).
The identified changes were useful in applying Descartes' Rule of Signs to deduce potential positive and negative zeros.

Complex Zeros

Complex zeros of a polynomial are solutions that are not real numbers. They typically appear in conjugate pairs when real coefficients are involved.

• Complex zeros include an imaginary part, denoted by \( i \), where \( i^2 = -1 \).
• If a polynomial has real coefficients, complex zeros occur in pairs like \( a + bi \) and \( a - bi \).

Given that our polynomial has a degree of 5, yet the number of real zeros is limited (either 2 or 0), the remaining zeros must be complex. By understanding that the total number of zeros, including complex ones, equals the polynomial's degree, we infer that any zeros not real must be complex.
This ensures that polynomials remain fully factored within the complex number system, respecting the degree.