Problem 24 - College Algebra

Question

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)=x^{3}+7 x^{2}+x+7 $$

Step-by-Step Solution

Answer

The function \(f(x)=x^{3}+7 x^{2}+x+7\) has an upper limit of zero positive real zeros and two negative real zeros.

Step 1: Identify Sign Changes for Positive Real Zeros

First, look at the given function \(f(x)=x^{3}+7 x^{2}+x+7\). The coefficients of the function are 1, 7, 1, and 7. There are no sign changes when going from one term to the next, therefore, according to Descartes's Rule of Signs, there are no positive real zeros for this function.

Step 2: Identify Sign Changes for Negative Real Zeros

Now, replace \(x\) with \(-x\) to evaluate the function for negative real zeros. The function becomes \(f(-x)=(-x)^{3}+7 (-x)^{2}+(-x)+7=-x^{3}+7 x^{2}-x+7\). The coefficients of this function are -1, 7, -1, and 7. There are exactly 2 sign changes (from -1 to 7 and from -1 to 7), which implies the function has a maximum of 2 negative real zeros.

Step 3: Formulate the Final Outcome

Based on the analysis of the function, the upper limit is zero positive real zeros and two negative real zeros. By subtracting multiples of 2 from the upper limit (because the actual number of roots could be less than the upper limit by a multiple of 2), we know that this function can have zero positive real zeros, and either 2 or 0 negative real zeros, depending on the specific nature of the function.

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving variables raised to various powers, known as degrees, along with their corresponding coefficients. A polynomial function is usually written in the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where \( n \) is a non-negative integer. Each term consists of a coefficient \( a_i \) and a variable \( x \) raised to the power \( i \).

Understanding the structure of polynomial functions is crucial, as they can model a wide variety of real-world scenarios, from physics to economics. They are defined for all real numbers and are continuous and smooth. Recognizing a polynomial function is the first step in analyzing its properties, among which is determining how many times it intersects the x-axis, known as its real zeros.

Real Zeros

The real zeros of a polynomial function are the points where the graph of the polynomial intersects the x-axis. For instance, real zeros correspond to the values of \( x \) that make \( f(x) = 0 \). A function's degree (the highest power of \( x \)) gives the maximum number of real zeros it can have.

Real zeros are important as they signify the solutions to the polynomial equation. They can be characterized by their multiplicity, which refers to the number of times a particular zero repeats. If a polynomial has distinct real roots, it means each zero occurs only once. For example, in the polynomial \( x^2 - 3x + 2 \), the real zeros are 1 and 2, since \( (x-1)(x-2) = 0 \) when \( x \) equals those values. Identifying real zeros helps in sketching the graph of the polynomial and understanding its various behaviors.

Sign Changes

Sign changes in a polynomial function take place when the sign of the coefficients changes from one term to the next as you move from left to right. Descartes's Rule of Signs provides a systematic approach to determine the number of positive or negative real zeros of a polynomial function by counting the number of sign changes.

To find the sign changes for positive real zeros, examine the polynomial as it is. Count the number of times the sign changes from positive to negative or vice versa. However, for detecting negative real zeros, substitute \(-x\) for \(x\) in the polynomial, which effectively flips the signs of all odd-powered terms. Then, look for sign changes in this new polynomial.

• If a polynomial has, for instance, 3 sign changes, it could have 3, 1, or 0 positive or negative real zeros, considering possible reductions by multiples of 2.

Understanding where and how these sign changes occur is crucial in applying Descartes's Rule of Signs effectively.

Positive and Negative Zeros

Positive and negative zeros of polynomial functions are categorized based on whether they are greater than or less than zero. Descartes's Rule of Signs cleverly utilizes the notion of sign changes to predict these values.

When looking for positive zeros, count the sign changes in the original polynomial. For negative zeros, replace every occurrence of \( x \) with \(-x\), and then count the sign changes in this new polynomial form. This technique doesn't just predict the number of zeros but also their nature.

• A polynomial might theoretically have the maximum number of positive or negative zeros equal to its degree, but typically, due to multiplicity and sign changes, the actual number is less.
• In the exercise example, you can see there are no sign changes in the original polynomial, indicating no positive zeros, while there are 2 sign changes in \( f(-x) \), suggesting the potential for 2 negative zeros.

This insight is foundational for anyone analyzing polynomials to find their real-world implications and uses.