Problem 70 - Precalculus Mathematics for Calculus

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. $$P(x)=x^{8}-x^{5}+x^{4}-x^{3}+x^{2}-x+1$$

Step-by-Step Solution

Answer

P(x) can have 5, 3, or 1 positive real zeros and no negative real zeros.

Step 1: Identify Sign Changes for Positive Real Zeros

To use Descartes' Rule of Signs, first count the number of sign changes in the polynomial \( P(x) = x^{8} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \). Observing the terms, the changes occur as follows: \(-x^{5} \to +x^{4} \), \(+x^{4} \to -x^{3} \), \(-x^{3} \to +x^{2} \), \(+x^{2} \to -x \), and \(-x \to +1\). This results in 5 sign changes.

Step 2: Determine Possible Positive Zeros

According to Descartes' Rule of Signs, the number of positive real roots is equal to the number of sign changes or less by an even integer. Hence, the number of possible positive real zeros is 5, 3, or 1.

Step 3: Identify Sign Changes for Negative Real Zeros

To determine sign changes for negative real zeros, substitute \( x = -y \) in the polynomial, giving \( P(-y) = (-y)^{8} - (-y)^{5} + (-y)^{4} - (-y)^{3} + (-y)^{2} - (-y) + 1 = y^8 + y^5 + y^4 + y^3 + y^2 + y + 1 \). This polynomial has no sign changes, indicating that there are no negative real zeros.

Step 4: Determine the Possible Total Number of Real Zeros

The possible number of positive real zeros are 5, 3, or 1. Since there are no negative real zeros, the possible total number of real zeros is either 5, 3, or 1.

Key Concepts

Polynomial Sign Changes

Descartes' Rule of Signs is a useful tool for predicting the number of real zeros of a polynomial. The first step involves identifying the number of sign changes in the polynomial's coefficients. A sign change occurs when consecutive coefficients switch from positive to negative or vice-versa.
This process is performed by examining the sequence of signs before each term. For the polynomial given by \( P(x) = x^{8} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \), observe the changes as follows:

• From \(-x^{5} \) to \(+x^{4} \)
• From \(+x^{4} \) to \(-x^{3} \)
• From \(-x^{3} \) to \(+x^{2} \)
• From \(+x^{2} \) to \(-x \)
• From \(-x \) to \(+1 \)

These transitions signify five sign changes, indicating the potential number of positive real zeros, as per Descartes' Rule.

Positive Real Zeros

Positive real zeros in a polynomial can be estimated through Descartes' Rule of Signs, which states that the number of positive real roots is equal to the number of sign changes in the polynomial or can be less by any even integer. By identifying five sign changes for our polynomial, we infer up to five possible positive real zeros.
However, the actual number could be 5, 3, or 1, considering that it might decrease by 2 (an even integer) each time. This logical deduction helps in estimating the possible real zeros before actually solving the polynomial, making it a powerful initial tool in algebra.

Negative Real Zeros

To find the negative real zeros, substitute \( x = -y \) in the original polynomial and compute the sign changes in \( P(-y) \). For our given polynomial:\(P(-y) = y^8 + y^5 + y^4 + y^3 + y^2 + y + 1\).
Observe that all coefficients are positive; hence, there are no sign changes. This implies there are no negative real zeros in this polynomial. Knowing that negative real zeros are absent can significantly narrow down the possibilities when sketching the graph or analyzing the roots.

Real Zeros of Polynomials

The total real zeros in a polynomial, according to Descartes' Rule, is closely tied to its degree and the sign changes. The degree of the polynomial \( P(x) = x^{8} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) is 8, but the relevant zeros are deduced from the rule.

• With 5, 3, or 1 positive real zeros potentially existing due to the sign changes
• No negative real zeros confirmed by substituting \(x = -y\)

This leads us to conclude the polynomial has either 5, 3, or 1 real zeros in total. Understanding these concepts is crucial when predicting the behavior of polynomials without graphing or solving them explicitly.