Problem 69 - 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 \(f\). $$f(x)=-2 x^{4}+13 x^{3}-21 x^{2}+2 x+8$$

Step-by-Step Solution

Answer

The real zeros of the function \(f(x) = -2 x^{4}+13 x^{3}-21 x^{2}+2 x+8\) are -1.5, 1, 3 and -2.

Step 1: Apply Descartes's Rule of Signs

Descartes's Rule of Signs is a method to determine the possible number of positive and negative real roots of a polynomial. The rule says that the number of positive real roots of a polynomial function is the same as the number of variations in sign among its coefficients, or less than that by a multiple of 2. Same method is applied for the number of negative roots by substituting \(x\) with \(-x\) in the function.\nOur function here is \(f(x)=-2 x^{4}+13 x^{3}-21 x^{2}+2 x+8\). The number of sign changes in \(f(x)\) is 2, so the possible numbers of positive real zeros are 2 or 0. Similarly, substitute \(x\) with \(-x\) to find the number of negative real zeros. \(f(-x) = -2(-x)^{4}+13(-x)^{3}-21(-x)^{2}+2(-x) +8 = -2x^4 -13x^3 -21x^2 -2x +8\), the number of sign changes is 4. So the possible numbers of negative real zeros are 4, 2 or 0.

Step 2: List the possible rational zeros

The Rational Zero Theorem states that if a polynomial has a rational zero, then it is of the form \(p/q\), where \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient.\nThe constant term in our function is 8 and the factors are ±1, ±2, ±4, ±8. The leading coefficient is -2, the factors are ±1, ±2. Hence, the possible rational zeros of the function are ±1, ±2, ±4, ±8, -1/2, 1/2, -2/2, 2/2, -4/2, 4/2, -8/2, 8/2.

Step 3: Graph the function

Using a graphing utility, the graph of function \(f(x)\) can be plotted. When the function crosses the x-axis, those are possible real roots. By noticing where the graph crosses the x-axis, it can be seen that roots are approximately near -1.5, 1, 3 and -2.

Step 4: Determine the real zeros

The real zeros of a function are the values of \(x\) for which \(f(x) = 0\). From the graph and the list of possible rational zeros, the estimated roots were -1.5, 1, 3 and -2. Now, it can be checked if these values satisfy the equation \(f(x)=0\), -2(-1.5)^4 +13(-1.5)^3 -21(-1.5)^2 +2(-1.5) + 8 = 0, -2(1)^4 +13(1)^3 -21(1)^2 +2(1) +8 = 0, -2(3)^4 +13(3)^3 -21(3)^2 +2(3) +8 = 0 and -2(-2)^4 +13(-2)^3 -21(-2)^2 +2(-2) +8 = 0. The values -1.5, 1, 3 and -2 are indeed the roots of the polynomial!

Key Concepts

Positive Real Zeros

Understanding the concept of positive real zeros in polynomial equations is crucial for grasping more complex mathematical concepts. In relation to Descartes's Rule of Signs, a 'positive real zero' is simply a root of the polynomial that is a positive number. When analyzing a polynomial, like our example function f(x)=-2x^4+13x^3-21x^2+2x+8, you'll want to determine how many times the coefficients change from positive to negative or vice versa. These sign changes indicate the maximum number of possible positive real zeros the polynomial might have. Additionally, this number can decrease by multiples of two, suggesting other kinds of roots, like complex ones, might exist. For the given function, the coefficient sign changes occur twice, indicating there could be two or zero positive real zeros.

When trying to predict these zeros, it's helpful to paint a mental picture that as a graph crosses the x-axis from negative to positive, it represents a positive real zero. This is where our graphing utilities come in handy to visually confirm our predictions, ensuring that the theoretical understanding aligns with the visual representation of the polynomial.

Negative Real Zeros

On the flip side, when we discuss 'negative real zeros', we're addressing roots that come out to be negative numbers. Applying Descartes's Rule of Signs requires a little twist here; we substitute our 'x' with a '-x' in the polynomial and then analyze the sign changes. For f(x)=-2x^4+13x^3-21x^2+2x+8, once we make our substitution, we end up with the transformed function f(-x) = -2x^4 -13x^3 -21x^2 -2x +8. This time, we count the sign changes in this new version of the function. The exercise showed four such changes, hinting at the possibility of four, two, or no negative real zeros.

This process provides a road map through the complex territory of negative roots. Using this rule gives us a starting point for experimentation and further validation using graphing techniques, where the negative zeros appear as the points where the graph snips through the x-axis into the negative zone.

Rational Zero Theorem

Moving on to a related topic, we encounter the 'Rational Zero Theorem', a guiding principle that helps us find the potential fraction form of polynomial roots. According to this theorem, if a polynomial has rational zeros, they're determined by the ratio of factors of the constant term to the factors of the leading coefficient. So, in the context of our polynomial f(x), the constant term is 8 ( with factors ±1, ±2, ±4, ±8) and the leading coefficient is -2 (with factors ±1, ±2). This translates to a list of potential rational zeros: ±1, ±2, ±4, ±8, ±1/2, and ±4/2.

The significance of the Rational Zero Theorem is profound as it narrows down the possible zeros to a finite list, making the task of finding actual zeros much more manageable. Then we use other methods, such as synthetic division or graph analysis, to check which of these potential zeros hold true for the polynomial in question.

Polynomial Roots

Lastly, the essence of our investigation often leads to finding the 'polynomial roots', the solutions to the equation f(x)=0. In simple terms, these are the x-values where the graph of the polynomial touches or crosses the x-axis. As shown in earlier steps, once potential zeros are hypothesized through Descartes's Rule of Signs and the Rational Zero Theorem, we can graph the function to visually observe these interceptions with the x-axis. It's like a detective lining up suspects (possible zeros) and using a lie detector (graphing utility) to find out who's telling the truth.

In our exercise, the estimated roots, spotted initially on the graph, were -1.5, 1, 3, and -2, which upon further scrutiny with the equation, indeed turned out to be the actual roots. Unraveling polynomial roots is a bit like untying a knotted string, requiring both methodical technique and a measure of finesse. This finesses comes with practice and a strong foundational understanding of the tools at hand—be it rules, theorems, or graphing technologies.