Problem 85 - 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 rational solutions are -1, 2, and 3.

Step 1: Identify Possible Rational Roots

According to the Rational Zeros Theorem, any rational solution of the polynomial equation \( 2x^4 - 5x^3 - 14x^2 + 5x + 12 = 0 \) must be a fraction \( \frac{p}{q} \), where \(p\) is a factor of the constant term (12) and \(q\) is a factor of the leading coefficient (2). The factors of 12 are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \), and the factors of 2 are \( \pm 1, \pm 2 \). Thus, the possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{7}{2} \).

Step 2: Test Possible Roots in Equation

Substitute each possible rational root into the polynomial equation to see which of them satisfy the equation by making the left-hand side equal to zero. This involves evaluating the polynomial for each potential root.

Step 3: Graph the Polynomial

Use graphing software or a graphing calculator to plot the polynomial \( y = 2x^4 - 5x^3 - 14x^2 + 5x + 12 \) in the viewing window \([-2, 5]\) by \([-40, 40]\). Look for the points where the graph intersects the x-axis, indicating the real rational roots.

Step 4: Identify the Rational Solutions

From the graph, identify the x-values where the curve crosses the x-axis. Verify these x-values against our testing from Step 2 to find exactly which of the possible roots are actual solutions.

Key Concepts

Polynomial Roots

In mathematics, polynomial roots are the values of the variable that make the polynomial equal to zero. This means if you substitute these values into the polynomial equation, they will satisfy it, resulting in a zero value. For example, in a polynomial like \(2x^4 - 5x^3 - 14x^2 + 5x + 12\), finding the roots means determining the values of \(x\) that solve the equation \(2x^4 - 5x^3 - 14x^2 + 5x + 12 = 0\).
These roots can be real or complex numbers. While complex roots come in conjugate pairs, real roots can either be rational or irrational. Rational roots are particularly convenient as they can be expressed as fractions of integers.
Understanding polynomial roots helps us graph these functions and understand their behavior, such as where they cross the x-axis, also known as the real roots.

Rational Solutions

Rational solutions of a polynomial are roots that can be expressed as fractions. These are numbers like \( \frac{1}{2}, 2, -3\), where both the numerator and the denominator are integers, and the denominator is not zero.
In the process of solving polynomial equations, identifying rational solutions can simplify your work significantly. For a polynomial equation like \(2x^4 - 5x^3 - 14x^2 + 5x + 12 = 0\), finding rational solutions involves using specific theorems and methods such as the Rational Roots Theorem.
Rational solutions help verify the accuracy of factoring, and they provide a clear view of which values should be plotted when graphing the polynomial. Students often find this part interesting yet challenging, as it involves testing extracted possible roots in the equation itself.

Graphing Polynomials

Graphing a polynomial function is a powerful way to visualize their behavior and understand their roots. By plotting a polynomial like \(y = 2x^4 - 5x^3 - 14x^2 + 5x + 12\) in a specified window, you can directly observe the points where the graph intersects the x-axis.
When the curve meets the x-axis, it's indicating the x-values that are roots of the polynomial, which can be rational solutions. For our polynomial, observing the window \([-2, 5]\) by \([-40, 40]\) allows us to see a comprehensive range of possible roots given the constraints.
Graphing not only helps confirm computed roots but also paints a picture of the overall shape of the polynomial—which is degree 4 in our example—and the symmetry or pattern it might exhibit.

Rational Roots Theorem

The Rational Roots Theorem is a key concept when dealing with polynomial equations. It asserts that for a given polynomial with integer coefficients, any rational root, expressed as \(\frac{p}{q}\), must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient.
For the polynomial \(2x^4 - 5x^3 - 14x^2 + 5x + 12\), the constant term is 12, with factors \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\), and the leading coefficient is 2, with factors \(\pm 1, \pm 2\). Therefore, the possible rational roots are combinations of these factors, such as 1, \(\frac{1}{2}\), -3, and so on.
This theorem significantly narrows down the number of potential rational solutions you need to test, saving time and effort while ensuring all possible options are considered.