Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of \(f\) (b) list the possible rational zeros of \(f,\) (c) use a graphing utility to graph \(f\) so that some of the possible zeros in parts (a) and (b) can be disregarded, and (d) determine all the real zeros of \(f\). $$f(x)=x^{3}+x^{2}-4 x-4$$

Step-by-Step Solution

Answer

Using Descartes's Rule of Signs, Rational Root theorem, and a graphing utility, we find that the real zeros of the function \(f(x) = x^{3} + x^{2} -4x - 4\) are -1, 1, and 2.

Step 1: Application of Descartes's Rule of Signs

Descartes's Rule of Signs is used to determine the possible number of positive and negative real zeros of a function. Count the number of sign changes in the coefficients of the function to determine the number of possible positive real zeros. The original function is \(f(x)= x^{3} + x^{2} - 4x - 4\). The coefficients are 1, 1, -4, -4. There are two sign changes, so there are either 2 or 0 positive real zeros. To find the negative zeros, replace \(x\) with \(-x\), which gives \(f(-x)= -x^{3} + x^{2} + 4x - 4\). There's only one change in signs, so there is exactly 1 negative real zero.

Step 2: List the possible rational zeros

To find the possible rational zeros of a function, one should use Rational Root Theorem. It says that if a function has a rational zero \(p/q\), then \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. For \(f(x)=x^{3} + x^{2} - 4x - 4\), the factors of the constant term -4 are ±1, ±2, ±4 and the factor of the leading coefficient 1 is ±1. So, the possible rational zeros of \(f(x)\) are ±1, ±2, ±4.

Step 3: Use a graphing utility

A graphing utility can be used to roughly determine the zeros of a function and to verify the answers obtained using mathematical methods. Use the graph of \(f(x)\), to eliminate some of the possible zeros derived from steps 1 and 2. You can see from the graph that the function intersects the x-axis at -2, 1, and -2. Eliminating others leaves us with the probable roots of -1, 1 and 2.

Step 4: Determine all real zeros

The last step involves confirming the real zeros derived from the graph by substituting the values into the function. After testing -1, 1 and 2, we see that these values all make the function zero. So, the real zeros of the function \(f(x) = x^{3} + x^{2} -4x - 4\) are -1, 1 and 2.

Key Concepts

Rational Root Theorem

Understanding the Rational Root Theorem is like having a mathematical spyglass that lets you zoom in on possible rational zeros of polynomial functions. This theorem provides a way to list all potential fractions (rational numbers) that could possibly be roots of the polynomial equation. If we think of a polynomial function represented as f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, where a_n and a₀ are the leading and constant coefficients respectively, then according to the Rational Root Theorem, any possible rational root in the form of p/q must have p as a factor of a₀ and q as a factor of a_n.

The refinement offered by the theorem is beneficial; it narrows down an infinitely large search to a manageable list of candidates. In our example, given f(x)=x^{3} + x^{2} - 4x - 4, the constant term is -4, and the leading coefficient is 1. Thus the possible rational zeros are combinations of factors of -4, which are ±1, ±2, and ±4, over the factors of 1, which is ±1. The Rational Root Theorem significantly reduces the trial and error process in finding the actual rational zeros of f(x).

Real Zeros of Polynomial Functions

The real zeros of polynomial functions are the x-values where the graph of the polynomial crosses or touches the x-axis. These values are also called roots or solutions, as they are the answers we often seek when solving polynomial equations. Discovering the real zeros is crucial because they represent the x-values for which the polynomial function equals zero, revealing much about the function’s behavior and graph.

When you look at a polynomial function, such as f(x)=x^{3} + x^{2} - 4x - 4, finding its real zeros involves examining where the function changes sign, which indicates that it passes through the x-axis. Using algebraic techniques like the Rational Root Theorem, factorization, or synthetic division, these values can often be determined. In our example, by applying Descartes's Rule of Signs, we predict the number of positive and negative real zeros and use the Rational Root Theorem to list possible rational zeros. Verifying these zeros algebraically or graphically ensures we've found all the real zeros of the function.

Graphing Utility to Find Zeros

A graphing utility is like a trusty sidekick for any student tackling polynomial functions. It visualizes what can sometimes be a complex function, making it easier to understand and analyze its behavior. To find zeros with a graphing utility, simply plot the function f(x) and look for the points at which the graph crosses the x-axis.

A well-designed graph could allow you to immediately spot the real zeros or at least approximate their values. In our example function f(x)= x^{3} + x^{2} - 4x - 4, graphing it helps us see that we can discard certain potential zeros from our list, enhancing our understanding of the function's behavior and zero locations. However, remember a graphing utility gives an approximation; it's essential to confirm any zeros you find by plugging them back into the equation to ensure they really do result in zero.