Problem 73 - Algebra and Trigonometry

Question

The equations in Exercises 72–75 have real roots that are rational. Use the Rational Zero Theorem to list all possible rational roots. Then graph the polynomial function in the given viewing rectangle to determine which possible rational roots are actual roots of the equation. $$ 6 x^{3}-19 x^{2}+16 x-4=0 ;[0,2,1] \text { by }[-3,2,1] $$

Step-by-Step Solution

Answer

After plotting, due to the varying factors dependent on the graphing tool used, the actual roots might vary. However, they will certainly be among the possible rational roots list compiled earlier: ±1, ±2, ±4, ±1/2, ±2/3, ±1/3, ±1/6.

Step 1: Apply Rational Zero Theorem

Use the Rational Zero Theorem, which states that if a polynomial has integer coefficients, then every rational zero will have the form \( p/q \) where \( p \) is a factor of the constant term (in this case, -4) and \( q \) is a factor of the leading coefficient (in this case, 6). The possible values for \( p \) are ±1, ±2, ±4, and the possible values for \( q \) are ±1, ±2, ±3, ±6. Hence, the possible rational roots are: ±1/1, ±2/1, ±4/1, ±1/2, ±2/2, ±4/2, ±1/3, ±2/3, ±4/3, ±1/6, ±2/6, and ±4/6.

Step 2: Reduce the Rational Values

Reduce the values into simplest form, eliminating duplicate values. As a result, the possible rational roots are: ±1, ±2, ±4, ±1/2, ±2/3, ±1/3, ±1/6.

Step 3: Graph the Polynomial Function

Graph the polynomial function \( 6x^{3}-19x^{2}+16x-4 \) in the viewing rectangle specified in the exercise, which is \([0,2,1] \) by \([-3,2,1]\).

Step 4: Identify the Actual Roots

After graphing the function, identify the x-intercepts, these values are the actual roots of the polynomial function. Depending on the graph, simply confirm which values amongst the possible rational roots listed in step 2 correspond to the x-intercepts (actual roots) of the plotted polynomial function.

Key Concepts

Polynomial Roots

When we talk about polynomial roots, we're referring to the solutions of a polynomial equation, or the values of x for which the polynomial equals zero. Understanding roots is vital because it not only helps in solving equations but also provides insight into the behavior of polynomial functions on a graph, such as where they intersect the x-axis. The roots of a polynomial are also called zeros or x-intercepts.

To find the roots of a polynomial like the one in the exercise, \(6x^3 - 19x^2 + 16x - 4 = 0\), one may employ various methods. Among these, factoring is a primary tool, but it is not always straightforward. This is where the Rational Zero Theorem can assist by narrowing down the list of possible rational roots.

Graphing Polynomial Functions

Graphing polynomial functions is a powerful way to visualize the behavior of polynomials and to find their roots. Before plotting the graph, it's helpful to know the possible number of x-intercepts and the end behavior of the polynomial, which depends on the degree and the leading coefficient. Using technology, one can graph a polynomial function within a viewing rectangle, which represents the range of values on the axes where the function's behavior will be observed.

Tips for Effective Graphing

• Choose an appropriate scale for both the x and y axes to capture all critical points.
• Identify the end behavior: as x approaches infinity, does the function rise or fall?
• Mark the possible rational roots on the graph, then draw the curve to see which ones are actual x-intercepts.

Rational Roots Identification

The method of rational roots identification starts with the Rational Zero Theorem. This theorem provides a systematic way of generating a list of candidate roots that are rational numbers. To narrow down this list to the actual roots, one can use various tests, including substituting candidates into the polynomial equation or utilizing tools like graphing calculators.

Here’s the catch though: the identified possible rational roots may not necessarily be the actual roots — they are merely candidates. To determine which are authentic, one might:

• Apply synthetic division or polynomial division to test each possible root.
• Use the graph of the polynomial function to observe which of the candidates are indeed the points where the function intersects the x-axis.

By examining the graph created in the previous step, you can see at a glance which of the possible roots make the polynomial equal to zero, thus identifying the actual roots of the equation.