Problem 91 - Algebra and Trigonometry

Question

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ 2 x^{4}-5 x^{3}-14 x^{2}+5 x+12=0 ; \quad[-2,5] \text { by }[-40,40] $$

Step-by-Step Solution

Answer

The real rational roots are -1, 2, and 3.

Step 1: Rational Zeros Theorem Application

According to the Rational Zeros Theorem, any possible rational root of a polynomial \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\) is of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\). For the polynomial \(2x^{4} - 5x^{3} - 14x^{2} + 5x + 12\), the constant term \(a_0 = 12\) and the leading coefficient \(a_n = 2\). The factors of 12 are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\), and the factors of 2 are \(\pm 1, \pm 2\). This gives us the possible rational roots \(\frac{p}{q} = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{7}{2}, \pm \frac{11}{2}\).

Step 2: Graphing the Polynomial

To visualize which of these potential roots are actually the real rational solutions, plot the polynomial \(2x^{4} - 5x^{3} - 14x^{2} + 5x + 12\) on a graph with the viewing window \([-2, 5]\) by \([-40, 40]\). The graph will intersect the x-axis at the points that correspond to the real roots of the polynomial.

Step 3: Identify the Real Rational Roots

Look at the intersections of the graph with the x-axis from the plot. The points of intersection within the viewing window identify the real rational roots of the equation. Comparing these x-intersects to the list of possible rational roots helps us confirm the actual rational roots.

Key Concepts

Polynomial Roots

To fully understand the concept of polynomial roots, think of them as the values which make the entire polynomial equation equal zero. In simple terms, if you have a polynomial equation and you plug in a number for the variable, if the equation balances to zero, then that number is a root. For example, with the polynomial presented in the exercise: \[ 2x^{4} - 5x^{3} - 14x^{2} + 5x + 12 = 0 \] we are tasked with finding the values of \(x\) that make this expression equal to zero. These values are crucial in algebra because they often help sketch the graph of the polynomial, solve equations, and understand the behavior of the polynomial over the set of real numbers.Finding these roots, especially for higher-degree polynomials, isn't always straightforward. In general, polynomial equations can have multiple roots, sometimes real, sometimes complex, and sometimes repeated or distinct. However, in this specific problem, we are only interested in the real and rational roots.

Rational Solutions

Rational solutions for polynomials are the roots that can be expressed as a simple fraction \(\frac{p}{q}\), where both \(p\) and \(q\) are integers and \(q eq 0\). This characteristic makes them appealing because they can be easily represented and used in further calculations.### Using the Rational Zeros Theorem The Rational Zeros Theorem is a fundamental tool to identify possible rational roots of a polynomial. It states that for a given polynomial expressed as \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), any potential rational zero (root) is a fraction where:

• \(p\) is a factor of the constant term \(a_0\)
• \(q\) is a factor of the leading coefficient \(a_n\)

In the exercise, the constant term \(a_0 = 12\) and the leading coefficient \(a_n = 2\). Listing all their factors and creating fractions \(\frac{p}{q}\), provides us with a list of possible rational solutions. This list narrows down the candidates we have to test to find actual roots.

Graphing Polynomials

Graphing polynomials is a powerful tool to visually identify their roots, especially when dealing with complex or high-degree polynomials. By plotting the polynomial function on a graph, the intersection points with the x-axis effectively show where the function equals zero, giving us a straightforward visual of the roots.When graphing, you often use a specific window to focus on the key areas where roots are likely. In the exercise, this is set from \([-2, 5]\) for the \(x\)-axis. The graph provides an immediate overview, highlighting intersections which correspond to real roots.Analyzing these intersections visually complements the rational solutions gathered from the Rational Zeros Theorem:

• It helps confirm which of those possibilities are actual roots.
• It aids in distinguishing between real and complex roots, as only real roots will intersect the \(x\)-axis.

Thus, combining graphing techniques with analytical methods paves the way for a clearer understanding and solution to polynomial equations.