Problem 2 - Precalculus Mathematics for Calculus

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; 0 negative real zeros.

Step 1: Introduction to Descartes' Rule of Signs

Descartes' Rule of Signs will help us determine the number of positive and negative real zeros of the polynomial function by analyzing the number of sign changes in the coefficients.

Step 2: Determine Positive Real Zeros

To find the number of positive real zeros, identify the number of sign changes in the coefficients of the polynomial \( P(x) = x^5 - 3x^4 + 2x^3 - x^2 + 8x - 8 \). The coefficients are \(1, -3, 2, -1, 8, -8\). The sign changes occur from \(1\rightarrow -3\), \(-3\rightarrow 2\), \(2\rightarrow -1\), and \(8\rightarrow -8\). Thus, there are 4 sign changes, indicating that there are 4, 2, or 0 positive real zeros.

Step 3: Replace x with -x for Negative Real Zeros

Substitute \( -x \) for \( x \) in the polynomial to obtain \( P(-x) = (-x)^5 - 3(-x)^4 + 2(-x)^3 - (-x)^2 + 8(-x) - 8 \), which simplifies to \(-x^5 - 3x^4 - 2x^3 - x^2 - 8x - 8\).

Step 4: Determine Negative Real Zeros

Examine the sign changes in the polynomial \( P(-x) = -x^5 - 3x^4 - 2x^3 - x^2 - 8x - 8 \). The coefficients \(-1, -3, -2, -1, -8, -8\) do not change signs at all, indicating that there are 0 negative real zeros.

Key Concepts

Understanding Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Specifically, a polynomial function in one variable, say \(x\), looks like this:

• \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\)

In this expression:

• \(n\) represents the degree of the polynomial, which is the highest power of \(x\) with a non-zero coefficient.
• \(a_n, a_{n-1}, \ldots, a_1, a_0\) are coefficients, which are usually real numbers.

Polynomial functions can have different shapes when graphed, influenced by their degree and the sign or value of coefficients. These functions are continuous and smooth, meaning they don't have sharp corners or breaks in their graphs, making them easier to study and analyze.

Identifying Real Zeros of Polynomial Functions

Real zeros or real roots of a polynomial function are the \(x\)-values where the polynomial evaluates to zero, \(P(x) = 0\). To find these zeros, one of the approaches we use is Descartes' Rule of Signs.

• This rule doesn't directly find the zeros but gives insight into how many positive and negative real zeros exist.

All real zer0s of a polynomial correspond to points where the graph crosses or touches the \(x\)-axis. To find the possible number of real zeros, polynomials can also be factored or solved using methods like the Rational Root Theorem, synthetic division, or graphing. Understanding Descartes' Rule of Signs provides a preliminary expectation of how many and what type of real zeros we should investigate.

Applying Descartes' Rule of Signs for Sign Changes

The key to using Descartes' Rule of Signs lies in counting the sign changes of the polynomial's coefficients.

• For positive real zeros, observe the original polynomial and count sign changes between consecutive coefficients.
• For negative real zeros, substitute \(-x\) for \(x\) and then count the changes.

In our exercise, with the polynomial \(P(x) = x^5 - 3x^4 + 2x^3 - x^2 + 8x - 8\), we noted:

• 4 sign changes in the original polynomial, predicting 4, 2, or 0 positive real zeros.
• When \(x\) is replaced by \(-x\), no sign changes occur, expecting 0 negative real zeros.

Remember that the number of predicted zeros can vary by multiples of 2 (e.g., 4, 2, or 0). This phenomenon is due to the squaring effect that roots have – pairs of complex conjugates can cancel out real sign changes, reducing the visible root count. Understanding this method helps in estimating the behavior and solution of polynomials accurately.