Problem 43 - Precalculus : Building Concepts and Connections

Question

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=4 x^{4}-5 x^{3}+6 x-3$$

Step-by-Step Solution

Answer

By Descartes' rule of signs, there are 2 or 0 positive real zeros and 1 negative real zero in the polynomial \(p(x) = 4x^4 - 5x^3 + 6x - 3\).

Step 1: Identify and count sign changes for positive real zeros

The given polynomial function is \(p(x) = 4x^4 - 5x^3 + 6x - 3\). Notice there are two sign changes: once from +4 to -5, and again from +6 to -3. Therefore, now that the number of sign changes in the polynomial equals 2, by Descartes' Rule of Signs, there will be exactly 2 or 0 positive real zeros.

Step 2: Identify and count sign changes for negative real zeros

The polynomial becomes \(p(-x) = 4(-x)^4 - 5(-x)^3 + 6(-x) - 3 = 4x^4 + 5x^3 - 6x - 3\). In this polynomial, you can see that there is 1 sign change, from +5 to -6, therefore there will be exactly 1 negative real zero.

Key Concepts

Polynomial Functions

A polynomial function is a type of expression composed of variables and coefficients, connected through operations such as addition, subtraction, multiplication, and non-negative integer exponents of variables. Polynomial functions are defined by their highest power, known as the degree. For instance, in the polynomial given, \(p(x) = 4x^4 - 5x^3 + 6x - 3\), we have a fourth-degree polynomial because the highest power of \(x\) is 4.

Polynomials can take many forms and have various degrees:

• A linear polynomial, like \(3x + 2\), has a degree of 1.
• A quadratic polynomial, such as \(x^2 - 4\), is degree 2.
• Higher-degree polynomials involve more complex behavior, with greater degrees indicating more turning points in their graphs.

Understanding the structure of polynomial functions helps in analyzing their behavior, such as where they cross the x-axis and how they change direction, known as turning points. These points and changes are directly connected to the concept of zeros, or solutions, to the polynomial equation \(p(x) = 0\).

Real Zeros

Real zeros of a polynomial function are the values of \(x\) for which the polynomial equals zero. They correspond to where the graph of the polynomial crosses or touches the x-axis. For example, if \(x = a\) is a real zero of \(p(x)\), then \(p(a) = 0\).

Finding real zeros is a critical part of analyzing a polynomial’s graph as it helps identify key points of intersection:

• They allow us to understand the roots and possible factorization of the polynomial.
• Real zeros inform us about the function's behavior between and beyond these points.

The number and type (positive, negative, or zero) of real zeros are sometimes predicted using Descartes' Rule of Signs. This rule helps determine potential zeros without explicitly solving the equation, providing a useful starting point for deeper algebraic analysis.

Sign Changes

Sign changes in a polynomial expression are essential in applying Descartes' Rule of Signs. This rule relates to counting the number of times the sign of the coefficients in the polynomial changes as you go from the term with the highest degree to a term with the lowest degree.

In the exercise, for the polynomial \(p(x) = 4x^4 - 5x^3 + 6x - 3\):

• The coefficients change from +4 to -5, creating the first sign change.
• They change again from +6 to -3, creating the second sign change.

Sign changes indicate the potential number of positive real zeros of the polynomial. To find potential negative real zeros, we replace \(x\) with \(-x\) and count the sign changes:

• In \(p(-x) = 4x^4 + 5x^3 - 6x - 3\), there is one sign change from +5 to -6.

Thus, the number of sign changes points to the possible number of positive and negative real zeros, offering a valuable tool for understanding polynomial behavior.