Problem 92 - 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.) $$ 3 x^{3}+8 x^{2}+5 x+2=0 ;[-3,3] \text { by }[-10,10] $$

Step-by-Step Solution

Answer

The rational root is \(x = -\frac{2}{3}\).

Step 1: Identifying Coefficients

First, we need to identify the coefficients of the polynomial. The given polynomial is \(3x^3 + 8x^2 + 5x + 2 = 0\). The leading coefficient (\(a\)) is 3, and the constant term (\(b\)) is 2.

Step 2: Applying the Rational Root Theorem

The Rational Root Theorem states that any rational solution of the polynomial equation \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0\) 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 \(a_n\). Here, \(p\) could be \(\pm 1, \pm 2\), and \(q\) could be \(\pm 1, \pm 3\).

Step 3: Listing Possible Rational Roots

Using \(p = \pm 1, \pm 2\) and \(q = \pm 1, \pm 3\), the possible rational roots are \(\pm 1, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3}\).

Step 4: Testing the Possible Roots

To determine which of these possible roots are actual roots of the polynomial, we substitute each into the polynomial \(3x^3 + 8x^2 + 5x + 2 = 0\). The roots need to satisfy the equation to be actual roots.

Step 5: Substituting and Solving

Let's substitute each possible rational root:- \(x = 1\): \(3(1)^3 + 8(1)^2 + 5(1) + 2 = 18\) (not a root)- \(x = -1\): \(3(-1)^3 + 8(-1)^2 + 5(-1) + 2 = 2\) (not a root)- \(x = 2\): \(3(2)^3 + 8(2)^2 + 5(2) + 2 = 62\) (not a root)- \(x = -2\): \(3(-2)^3 + 8(-2)^2 + 5(-2) + 2 = -12\) (not a root)- \(x = \frac{1}{3}\): \(3(\frac{1}{3})^3 + 8(\frac{1}{3})^2 + 5(\frac{1}{3}) + 2 = 2.70\) (not a root)- \(x = -\frac{1}{3}\): \(3(-\frac{1}{3})^3 + 8(-\frac{1}{3})^2 + 5(-\frac{1}{3}) + 2 = 3\) (not a root)- \(x = \frac{2}{3}\): \(3(\frac{2}{3})^3 + 8(\frac{2}{3})^2 + 5(\frac{2}{3}) + 2 = 5.25\) (not a root)- \(x = -\frac{2}{3}\): \(3(-\frac{2}{3})^3 + 8(-\frac{2}{3})^2 + 5(-\frac{2}{3}) + 2 = 0\) (root)\.

Step 6: Graphing the Polynomial

To confirm and visualize the roots, plot the polynomial \(3x^3 + 8x^2 + 5x + 2\) in the given viewing rectangle \([-3, 3] \text{ by } [-10, 10]\). By examining the graph, \(-\frac{2}{3}\) is verified as a rational root and matches the intercepts seen on the graph.

Key Concepts

Polynomial roots

Polynomial roots are the solutions or values of the variable that satisfy the polynomial equation. For example, in the given polynomial \(3x^3 + 8x^2 + 5x + 2 = 0\), finding the roots means finding the \(x\)-values that make the polynomial equal zero. Polynomials can have multiple roots, and they may be real or complex numbers. Real roots can also be either rational or irrational. In this exercise, we specifically look for real solutions. The real roots of a polynomial can be visualized graphically as the points where the polynomial's graph crosses the x-axis. Understanding polynomial roots helps in various mathematical applications, ensuring better insight into the behavior of polynomial equations.

Rational solutions

Rational solutions refer to the values that solve a polynomial equation and can be expressed as a fraction of two integers \(\frac{p}{q}\), where both \(p\) and \(q\) are integers and \(q eq 0\). The Rational Root Theorem is a useful tool for finding possible rational roots of polynomial equations.

Using the Rational Root Theorem

A rational root \(\frac{p}{q}\) must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient. This allows us to generate a list of potential rational solutions.

• For the polynomial \(3x^3 + 8x^2 + 5x + 2\), the constant term is 2 and the leading coefficient is 3.
• The potential factors of 2 are \(\pm 1, \pm 2\) and the potential factors of 3 are \(\pm 1, \pm 3\), leading to possible rational roots: \(\pm 1, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3}\).

All candidates must be tested in the polynomial to see if they satisfy the equation \(3x^3 + 8x^2 + 5x + 2 = 0\) to determine if they are actual rational solutions.

Graphing polynomials

Graphing polynomials is a powerful method that visually illustrates the function's behavior, helping to confirm the existence and nature of its roots. By examining the polynomial \(3x^3 + 8x^2 + 5x + 2\), we can utilize a graph to validate our findings from the Rational Root Theorem.When graphing, we plot the function over a specific viewing range. In this example, the graphing window was set from \([-3, 3]\) on the x-axis and \([-10, 10]\) on the y-axis. This allows us to observe where the curve intersects the x-axis, which indicates the roots.

Why Graphing Helps

Graphing provides a visual check of:

• The number of real roots — where the polynomial crosses the x-axis.
• The approximate value of the roots — showing precise locations of x-intercepts.
• Confirmation of calculated roots — such as confirming that \(x = -\frac{2}{3}\) is indeed a solution as seen by a crossing point.

Using graphs alongside algebraic methods is a comprehensive way to solve and understand polynomial equations.