Problem 69 - Algebra and Trigonometry

Question

By the Zeros Theorem, every nth-degree polynomial equation has exactly n solutions (including possibly some that are repeated). Some of these may be real, and some may be imaginary. Use a graphing device to determine how many real and imaginary solutions each equation has. (a) \(x^{4}-2 x^{3}-11 x^{2}+12 x=0\) (b) \(x^{4}-2 x^{3}-11 x^{2}+12 x-5=0\) (c) \(x^{4}-2 x^{3}-11 x^{2}+12 x+40=0\)

Step-by-Step Solution

Answer

(a) 3 real, 1 imaginary (b) 0 real, 4 imaginary (c) 2 real, 2 imaginary

Step 1: Identify the Degree of the Polynomial

For each equation, identify the degree of the polynomial. Each equation is of degree 4, therefore according to the Zeros Theorem, each one will have 4 solutions in total.

Step 2: Use a Graphing Device

Graph each polynomial equation using a graphing calculator or graphing software. Identify the x-intercepts of each graph, which represent the real solutions.

Step 3: Analyze Graph for Real Solutions (Part a)

For the equation \(x^{4}-2x^{3}-11x^{2}+12x=0\), the graph shows 3 distinct x-intercepts at approximately \(x=0\), \(x=1.5\), and \(-1.5\). This indicates there are 3 real solutions.

Step 4: Determine Imaginary Solutions (Part a)

Since there are a total of 4 solutions, and 3 of them are real, the equation must have 1 imaginary solution.

Step 5: Analyze Graph for Real Solutions (Part b)

For the equation \(x^{4}-2x^{3}-11x^{2}+12x-5=0\), the graph shows no x-intercepts. This indicates there are no real solutions.

Step 6: Determine Imaginary Solutions (Part b)

Since the equation has a degree of 4 and no real solutions, there must be 4 imaginary solutions.

Step 7: Analyze Graph for Real Solutions (Part c)

For the equation \(x^{4}-2x^{3}-11x^{2}+12x+40=0\), the graph shows two x-intercepts. This indicates there are 2 real solutions.

Step 8: Determine Imaginary Solutions (Part c)

Because the equation is of degree 4, and 2 solutions are real, there must be 2 imaginary solutions.

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions that involve a sum of powers of a variable. Each power, also called a term, is multiplied by a coefficient. For example, the equation \(x^4 - 2x^3 - 11x^2 + 12x = 0\) is a fourth-degree polynomial. The highest power, which is 4 in this case, determines the degree of the polynomial. Polynomial equations can have many forms and degrees, and solving them means finding the values of the variable that make the equation true. These solutions are known as the roots of the equation. The number of solutions corresponds to the degree of the polynomial as per the Zeros Theorem. This theorem states that an nth degree polynomial has exactly n roots, although some may be repeated.

Real and Imaginary Solutions

When solving polynomial equations, you will encounter real and imaginary solutions. Real solutions are the values of the variable that you can plot on the x-axis of a graph – they can be positive, negative, or zero. These solutions manifest as x-intercepts on the graph of the polynomial equation.Imaginary solutions, however, do not correspond to points on the real axis. They involve the imaginary unit \(i\), where \(i^2 = -1\). Imaginary roots happen in pairs when dealing with polynomials with real coefficients. For example, if you find a non-real solution, there is usually a conjugate imaginary solution.

Graphing Calculators

Graphing calculators are valuable tools for visually understanding polynomial equations. They allow you to input your equation and produce a graph that helps identify the real solutions of the polynomial – these are seen as x-intercepts on the graph. With a graphing calculator, you can easily determine how many real solutions exist. If an equation has no x-intercepts, it means all solutions are imaginary. By visually analyzing these graphs, you can infer complex relationships without manual calculations, saving time and reducing errors. In exercises like those provided, graphing calculators can instantly reveal whether you are dealing with 0, 2, or more real solutions, thereby also helping deduce the imaginary ones due to the total number of solutions being known from the polynomial's degree.

Degree of Polynomial

The degree of a polynomial is critical in understanding the number of solutions it can possess. Defined as the highest exponent of the variable in the polynomial, the degree also directly relates to the possible number of solutions (or roots) of the equation. For instance, a fourth-degree polynomial will have exactly four solutions in total due to the Zeros Theorem. These can be a mix of real and imaginary ones. As seen in the example equations, each equation is degree 4, predicting a total of four solutions regardless of their nature. This understanding allows us to approach the equation strategically with the goal of uncovering all possible solutions.