Problem 2 - Algebra and Trigonometry

Question

Using Descartes' Rule of Signs, we can tell that the polynomial \(P(x)=x^{5}-3 x^{4}+2 x^{3}-x^{2}+8 x-8\) has _________, _________, or __________positive real zeros and _________ negative real zeros.

Step-by-Step Solution

Answer

4, 2, or 0 positive real zeros and 0 negative real zeros.

Step 1: Identify the Polynomial and Its Terms

The polynomial is given as \( P(x) = x^5 - 3x^4 + 2x^3 - x^2 + 8x - 8 \). We need to determine the number of sign changes for both positive and negative real zeros according to Descartes' Rule of Signs.

Step 2: Apply Descartes' Rule of Signs for Positive Zeros

Descartes' Rule of Signs states that the number of positive real zeros is either equal to the number of sign changes between consecutive terms of \( P(x) \) or less by an even number. Examine the sequence of coefficients: \(+1, -3, +2, -1, +8, -8\). There are four sign changes: from \( +1 \) to \( -3 \), \( -3 \) to \( +2 \), \( +2 \) to \( -1 \), and \( -1 \) to \( +8 \). Hence, there can be 4, 2, or 0 positive real zeros.

Step 3: Apply Descartes' Rule of Signs for Negative Zeros

To find the number of negative real zeros, substitute \( -x \) for \( x \) in \( P(x) \) to get \( P(-x) = (-x)^5 - 3(-x)^4 + 2(-x)^3 - (-x)^2 + 8(-x) - 8 \), which simplifies to \(-x^5 - 3x^4 - 2x^3 - x^2 - 8x - 8 \). The sequence of signs is \(-, -, -, -, -, -\), which has no sign changes, meaning there are 0 negative real zeros.

Key Concepts

Polynomial

A polynomial is a type of mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, it's a series of numbers and variables combined by addition or subtraction, often written in terms of a single variable like, for instance, “x”.
For example, the polynomial provided in the exercise is:

• The variable is "x".
• The highest power of x, which represents the degree of the polynomial, is 5 (since it is a fifth-degree polynomial).
• The terms are written from the highest degree to the lowest degree: \(x^5 - 3x^4 + 2x^3 - x^2 + 8x - 8\).

Polynomials are essential in algebra because they can be used to express a wide range of real-world phenomena. In this context, understanding the structure of a polynomial is crucial since it sets the stage for applying techniques like Descartes' Rule of Signs to analyze the nature of its roots.

Positive Real Zeros

Examining the "positive real zeros" means we focus on the roots of the polynomial that are positive numbers. A real zero is essentially a solution to the equation formed when the polynomial equals zero.
To find the possible number of positive real zeros of the polynomial \(P(x)=x^{5}-3x^{4}+2x^{3}-x^{2}+8x-8\), we use Descartes' Rule of Signs. This rule helps us predict how many positive real zeros there might be based on the sequence of coefficients.
The steps are as follows:

• Look at the signs of each coefficient in the polynomial: \(+1\), \(-3\), \(+2\), \(-1\), \(+8\), \(-8\).
• Count how many times the sign changes from one term to the next. In this polynomial, the sign changes four times.

According to Descartes' Rule of Signs, the number of positive real zeros is the number of sign changes or less by an even number. Thus, in this case, the polynomial can have 4, 2, or 0 positive real zeros.

Negative Real Zeros

When talking about "negative real zeros," we mean the values of "x" that make the polynomial equal to zero, and these values are negative numbers.
To use Descartes' Rule of Signs to find negative real zeros, simply substitute \(-x\) for \(x\) in the polynomial. Then, check the sequence of signs in the new polynomial:\[P(-x) = (-x)^5 - 3(-x)^4 + 2(-x)^3 - (-x)^2 + 8(-x) - 8 \]This simplifies to:\[-x^5 - 3x^4 - 2x^3 - x^2 - 8x - 8\]

• Notice that all terms have negative coefficients: \(-, -, -, -, -, -\).
• There are no sign changes throughout the sequence.

No sign changes imply there are zero negative real zeros for this polynomial. Therefore, when following Descartes' Rule of Signs, it becomes straightforward to conclude the number of negative real roots a polynomial can have. Understanding this helps in predicting the behavior of the polynomial without actually solving the equation.