Problem 34 - 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. $$ f(x)-x^{3}+7 x^{2}+x+7 $$

Step-by-Step Solution

Answer

So, there are possible 0 positive real zeros and 2 (or 0) negative real zeros.

Step 1: Identify the terms and their coefficients

in the function \( f(x) = x^{3} + 7x^{2} + x + 7 \), the coefficient of each term are: +1 (for \(x^3\)), +7 (for \(7x^2\)), +1 (for \(x\)), and+7 (for constant term 7).

Step 2: Apply Descartes's rule for positive zeros

Find the number of sign changes from term to term. As all the coefficients are positive, there's no sign change indicating that there are 0 positive real zeros.

Step 3: Substitute x by -x to find negative zeros

We substitute \(x\) by \(-x\) to get the function \( f(-x) = -x^3 + 7x^2 - x + 7 \).

Step 4: Apply Descartes's rule for negative zeros

Checking the signs for \( f(-x) \), the coefficients now are: -1 (for \(x^3\)), +7 (for \(7x^2\)), -1 (for \(x\)), and +7 (for constant term 7). We see two sign changes: from -1 to +7 and from -1 to +7. Therefore, the function could have 2 or 0 (2 minus multiples of 2) negative real zeros.

Key Concepts

Positive Real Zeros

Positive real zeros are the solutions to a polynomial equation where the variable, typically denoted as \(x\), is positive. These solutions, or roots, are valuable because they tell us where the graph of the polynomial function intersects the x-axis on the positive side. To determine the number of positive real zeros using Descartes's Rule of Signs, we observe the polynomial's sign changes. In the original exercise, the function given was \( f(x) = x^3 + 7x^2 + x + 7 \). Here, all coefficients are positive, which means there are no sign changes between the terms. Consequently, Descartes's Rule of Signs indicates that there are no positive real zeros for this polynomial function. This is an essential first step in understanding the roots of the polynomial and how they impact the overall shape of the graph.

Negative Real Zeros

Negative real zeros are solutions where the variable \( x \) is negative. These zeros identify where the graph of the polynomial function intersects the x-axis on the negative side. To find negative real zeros using Descartes's Rule of Signs, we substitute \( x \) with \( -x \) in the polynomial expression. By doing this transformation for the given function, the polynomial becomes \( f(-x) = -x^3 + 7x^2 - x + 7 \). Now, observe the sequence of coefficients: -1, +7, -1, and +7. The rule states that the number of negative real zeros corresponds to the number of sign changes in this new sequence. Here we notice two changes: from -1 to +7 and from -1 to +7. Hence, the polynomial could have either 2 negative real zeros or 0 negative real zeros (since we consider all numbers of zeros reduced by even multiples). Understanding these potential zeros helps anticipate where the graph might cross the x-axis on the negative side.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of one or more variables multiplied by coefficients. They can provide lots of information about their graphical representation through their structure. Polynomial functions range from simple linear functions, like \( f(x) = x + 1 \), to complex multi-variable equations. They are foundational in algebra and calculus because they are continuous and easy to differentiate. The degree of the polynomial, which is the highest power of the variable, usually indicates the number of possible roots or intersections with the x-axis, though not all roots might be real. In the exercise, the polynomial is a cubic one, meaning it has a degree of three. Thus, it could have up to three real roots. However, not all degrees necessarily translate to visible zeros, as some roots can be complex (not intersecting the x-axis). Understanding the nature of polynomial functions, including their possible zeros, informs how algebraic techniques, such as Descartes's Rule of Signs, can uncover insights about their graphical behavior.