Problem 70 - 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 \(f\). $$f(x)=4 x^{4}-17 x^{2}+4$$

Step-by-Step Solution

Answer

The possible numbers of positive and negative real zeros of \(f\) are 0 and 1 respectively. The possible rational zeros of \(f\) are \(\pm1, \pm2, \pm4\). After graphing the function, -1, -4, 1, and 4 can be disregarded. Finally, the real zeros of \(f\) are -2, 0, and 2.

Step 1:.Apply Descartes's Rule of Signs

The function \(f(x)=4 x^{4}-17 x^{2}+4\) has no sign changes, so it has no positive real zeros. For \(f(-x)\), the sign changes once, so there is exactly one negative real zero.

Step 2: List the Possible Rational Zeros Using Rational Root Theorem

According to the Rational Root Theorem, the possible rational zeros are \(\pm \frac{p}{q}\), where \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient. In this case, the possible rational zeros are \(\pm1, \pm2, \pm4\).

Step 3: Use a Graphing Utility to Graph the Function

When the function \(f(x)=4 x^{4}-17 x^{2}+4\) is graphed, it shows that the function intersects the x-axis at -2, 0, and 2. So the possible zeros -1, -4, 1, and 4 can be disregarded.

Step 4: Determine All the Real Zeros of the Function

By substituting the possible rational zeros into the function, it can be seen that \(f(-2)=0, f(0)=0, f(2)=0\). So, the real zeros of the function are -2, 0, and 2.

Key Concepts

Descartes's Rule of Signs

Descartes's Rule of Signs is a powerful mathematical tool for predicting the number of positive and negative real zeros in a polynomial function. Starting with a given polynomial, you count the number of times the coefficients change sign. For our exercise, the polynomial \(f(x)=4 x^{4}-17 x^{2}+4\) doesn’t change signs, which suggests zero positive real zeros. However, when we replace \(x\) with \(-x\), we observe one sign change, indicating exactly one negative real zero. This information narrows down the search for real zeros significantly and helps in sketching the rough graph of the polynomial.

The usefulness of this rule lies in its simplicity for a quick analysis of the possible nature of zeros without actual calculation. It is essential, however, to remember that it gives the maximum number of real zeros in terms of positive and negatives, which could include possible combinations with even numbers of non-real complex zeros. For instance, if a polynomial has two sign changes, it might have two or zero positive real zeros.

Rational Root Theorem

The Rational Root Theorem is a handy guideline when dealing with polynomials. It allows you to list all the possible rational zeros that a polynomial function could have. It states that if a polynomial has any rational zeros, they are in the form of \(\pm \frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. In our exercise, the constant term is 4 and the leading coefficient is also 4, which leads to potential rational zeros of \(\pm1, \pm2, \pm4\).

This theorem does not guarantee that all listed numbers will be zeros, but it's a starting point for further investigation, like substitution or graphing. To narrow down our list, we match these potential zeros against the number of positive and negative real zeros predicted by Descartes's Rule of Signs and further verify them through graphing or synthetic division.

Graphing of Polynomials

Graphing of polynomials is a visual approach to understanding the behavior of these functions. By plotting the function on a coordinate plane, one can often identify the zeros of the function—points where the graph crosses the x-axis. For the polynomial \(f(x)=4 x^{4}-17 x^{2}+4\), graphing reveals intersections at -2, 0, and 2, confirming these as real zeros. We can disregard some possible zeros from the Rational Root Theorem, as they do not correspond to these intersections.

The graph provides a clear, intuitive picture of the function's behavior. It helps confirm findings from algebraic methods such as Descartes's Rule of Signs and the Rational Root Theorem, and it can unveil other important characteristics like turning points, end behavior, and symmetry. Remember, graphing utilities and calculators serve as a valuable check against algebraic solutions but should not replace conceptual understanding.