Problem 91 - Precalculus Mathematics for Calculus

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. $$2 x^{4}-5 x^{3}-14 x^{2}+5 x+12=0 ; \quad[-2,5] \text { by }[-40,40]$$

Step-by-Step Solution

Answer

The rational roots are determined using the Rational Root Theorem and verified using graphing; the actual rational solutions are those that correspond to x-intercepts on the graph.

Step 1: Identifying the Coefficients

For the polynomial \(2x^4 - 5x^3 - 14x^2 + 5x + 12\), identify the coefficients. The constant term is 12, and the leading coefficient is 2.

Step 2: Applying the Rational Root Theorem

The Rational Root Theorem states that any rational solution, in the form of a fraction \( p/q \), is such that \( p \) divides the constant term and \( q \) divides the leading coefficient. List all divisors of 12: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \) and of 2: \( \pm 1, \pm 2 \).

Step 3: Listing Possible Rational Roots

From the Rational Root Theorem, the possible rational roots are the combinations of divisors: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

Step 4: Graphing the Polynomial

Use a graphing calculator or software to graph the polynomial \(2 x^{4}-5 x^{3}-14 x^{2}+5 x+12=0\) in the specified viewing window \([-2,5]\) by \([-40,40]\). Observing the graph helps in identifying actual roots among the potential rational solutions.

Step 5: Verifying Real Rational Roots

Compare the intercepts from the graph with potential rational roots. Check each potential solution by substituting in the original polynomial equation to verify.

Key Concepts

Polynomial Roots

When we talk about polynomial roots, we are referring to the values of \( x \) that make the polynomial equation equal to zero. Imagine a graph of the polynomial equation; the points where the graph crosses the x-axis are the roots.
For the polynomial \(2x^4 - 5x^3 - 14x^2 + 5x + 12\), finding the roots means discovering which \( x \)-values satisfy \(2x^4 - 5x^3 - 14x^2 + 5x + 12 = 0\). These roots can be real or complex numbers, but in this exercise, we focus on real rational roots as specified in the problem.
Rational roots are numbers that can be expressed as fractions \( \frac{p}{q} \), where both \( p \) and \( q \) are integers, and the denominator \( q \) is not zero. Identifying these roots is crucial to understanding the behavior of the polynomial.

Graphing Polynomials

Graphing a polynomial helps visualize where the polynomial equals zero, meaning it identifies the roots. By plotting the polynomial \(2 x^{4}-5 x^{3}-14 x^{2}+5 x+12=0\) within the viewing window \([-2,5] \) by \([-40,40] \), students can see the behavior of the graph across this interval.
The graph's intercepts with the x-axis show potential rational roots. These x-intercepts occur where the polynomial value is zero. Thus, graphing is an essential step to confirm which values from our list of possible rational roots are genuine solutions.
A graphing calculator or computer software is the best tool for this task, as it accurately plots the curve and marks the intercepts, giving clear evidence of the roots. This visual representation can make understanding the polynomial's solution set more intuitive.

Coefficients of Polynomial

In any polynomial such as \(2x^4 - 5x^3 - 14x^2 + 5x + 12\), the coefficients play a vital role. These are the numbers attached to each term that influence the equation's value. In our example, 2 is the leading coefficient corresponding to the highest power term \(x^4\), and 12 is the constant term without any attached \(x\).
The coefficients are crucial when using the Rational Root Theorem. They help determine the factors to consider when listing all possible rational roots.

• The leading coefficient (2) influences the potential denominators in our list of possible roots.
• The constant coefficient (12) influences the potential numerators.

By carefully examining these coefficients, we accurately apply the Rational Root Theorem to find rational solutions.

Possible Rational Roots

The Rational Root Theorem allows us to identify possible rational roots by using the coefficients we've determined. The theorem suggests that if \( \frac{p}{q} \) is a rational root, \( p \) must be a factor of the constant term, and \( q \) must be a factor of the leading coefficient.
In our polynomial:

• Possible values for \( p \) (divisors of 12) include: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).
• Possible values for \( q \) (divisors of 2) include: \( \pm 1, \pm 2 \).

From these, we combine to form potential rational roots, such as \( \pm 1, \pm 2, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \) and so on.
Each possible root should be tested in the original polynomial equation. If the polynomial equals zero for a given root, it’s verified as an actual root. This method saves time by narrowing down candidates before computational or graphical verification.