Problem 36 - College Algebra

Question

In Exercises \(33-38,\) use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)--2 x^{3}+x^{2}-x+7 $$

Step-by-Step Solution

Answer

The function \(f(x) = -2x^3 + x^2 - x + 7\) could have either 3 or 1 positive real roots, and it could have 1 or no negative real roots.

Step 1: Identify the coefficients of the polynomial function

Look at the polynomial and identify the coefficients. The coefficients in this case are -2, 1, -1, and 7.

Step 2: Determine the number of sign changes for possible positive roots.

Count the number of sign changes in the sequence of coefficients. In this case, there are 3 sign changes (-2 to 1, 1 to -1, -1 to 7). Therefore, there could be 3, or 3-2k positive real roots. As k is natural, possible positive real roots are 3 or 1.

Step 3: Determine the number of sign changes for possible negative roots

Now for negative roots, replace x by -x in the function, it becomes \(f(-x)=-2(-x)^3+(-x)^2--x+7=x^3+x^2+x+7\). The change of signs in sequence of coefficients here is 1 (from 1 to -1). Therefore, there could be 1, or 1-2k negative real roots. As k is natural, possible negative real roots are 1 or 0.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of a variable, usually written as a combination of coefficients and the variable raised to various powers. The general form of a polynomial function can be expressed as: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]Here, \(a_n, a_{n-1}, \ldots, a_0\) are constants known as coefficients, and \(n\) represents the degree of the polynomial, which is the highest power of \(x\). In the given exercise, the polynomial function is \(f(x) = -2x^3 + x^2 - x + 7\), which is a cubic polynomial because the highest power of \(x\) is 3.

• The degree of the polynomial tells us the maximum number of roots we can expect.
• Each power of \(x\) in a polynomial represents a term.
• Polynomial functions can graph as smooth, continuous curves in a coordinate system.

Understanding the structure of polynomial functions is crucial for analyzing and predicting their behavior, especially when applying rules like Descartes's Rule of Signs to explore the nature of roots.

Real Zeros

The roots or zeros of a polynomial function are the values of \(x\) that make the function equal to zero, i.e., \(f(x) = 0\). Real zeros are simply these root values that are real numbers.

• They represent the points where the graph of the polynomial crosses the x-axis.
• Finding real zeros can give insight into the behavior and characteristics of the polynomial.

To determine the real zeros from Descartes's Rule of Signs, we primarily focus on counting sign changes in the polynomial's coefficients. Each sign change indicates a possible real zero, though not all may be distinct because roots can also have multiplicities.

• In our polynomial \(-2x^3 + x^2 - x + 7\), we're interested in identifying the number of potential positive or negative real zeros.

Analyzing real zeros helps in comprehending the graphical representation of the polynomial and solving equations involving polynomials.

Sign Changes

Sign changes refer to the transition between positive and negative coefficients in a polynomial function, or between negative and positive, as you move from one term to the next. This is crucial for using Descartes's Rule of Signs.When you list the coefficients of the polynomial \(-2, 1, -1, 7\), a sign change occurs whenever two consecutive coefficients switch from positive to negative or vice versa.

• For positive roots, count the sign changes directly in the polynomial: \(-2 \rightarrow 1\), \(1 \rightarrow -1\), and \(-1 \rightarrow 7\), amounting to three sign changes.
• For negative roots, substitute \(-x\) into the polynomial, transforming it into \(f(-x) = -2(-x)^3 + (-x)^2 - (-x) + 7 = x^3 + x^2 + x + 7\). This has one sign change \(1 \rightarrow -1\).

Tracking sign changes helps estimate the number of possible positive or negative real roots accurately.

Positive and Negative Roots

Using Descartes's Rule of Signs, we can predict the possible number of positive and negative roots of a polynomial.

• Positive roots are found by counting the sign changes from the original polynomial.
• Negative roots are determined by substituting \(-x\) into the function and then counting the sign changes in the resulting polynomial.

In the given polynomial \(-2x^3 + x^2 - x + 7\):

• There are three sign changes for positive roots, suggesting there could be 3 or 1 positive real roots.
• After substituting \(-x\), there is one sign change for negative roots, suggesting there could be 1 or 0 negative real roots.

This rule does not provide exact root counts but presents potential root numbers. This is because the actual number of roots can be less by an even number if complex roots are involved. Using this method simplifies assessing root possibilities without solving the polynomial directly.