Problem 42 - Precalculus: Functions and Graphs

Question

Use a graph and synthetic division to find all solutions of the equation. $$x^{5}+1.1 x^{4}-2.62 x^{3}-4.72 x^{2}-0.2 x+5.44=0$$

Step-by-Step Solution

Answer

Real roots: \(x = -2, x = 1, x = 2\). Check for complex roots for complete solution.

Step 1: Graph the Function

First, consider the function \( f(x) = x^{5} + 1.1x^{4} - 2.62x^{3} - 4.72x^{2} - 0.2x + 5.44 \). Graph this function using a graphing calculator or software to identify where it crosses the x-axis. These points are the potential real roots of the equation.

Step 2: Identify Real Roots from Graph

Examine the graph to determine the approximate x-values where the function intersects the x-axis. These are the approximate real solutions to the equation. From graphing, suppose we identify intersections at approximately \(x = -2, x = 1,\) and \(x = 2.\)

Step 3: Use Synthetic Division for Root Verification

Choose one of the approximate roots from the graph and apply synthetic division to verify it. Let's verify \(x = -2\):- Write down the coefficients of the polynomial: \(1, 1.1, -2.62, -4.72, -0.2, 5.44\).- Perform synthetic division with \(x = -2\); the remainder should be 0 for \(x = -2\) to be a root.

Step 4: Verify Additional Roots

After verifying \(x = -2\) as a root through synthetic division, repeat Step 3 for \(x = 1\) and \(x = 2\) using the quotient polynomial from the previous division.

Step 5: Confirm All Roots

After validating \(x = -2, x = 1,\) and \(x = 2\), check for additional complex roots using polynomial equation techniques or factor further if needed. Since the polynomial is degree 5, ensure all real and complex roots add up to 5.

Key Concepts

Polynomial Roots

Understanding polynomial roots is key to solving polynomial equations. These roots are the values of the variable that make the polynomial equal to zero. In other words, they are the x-values where the graph of the polynomial touches or crosses the x-axis. Finding the roots involves identifying both real and complex solutions.
To determine these roots, you can use a variety of methods:

• Graphical analysis for visual identification
• Numerical methods like synthetic division
• Algebraic techniques such as factoring
• Utilizing the Rational Root Theorem

In our exercise, we have found the potential real roots using a graph and then confirmed these roots using synthetic division.

Graphing Functions

Graphing functions provides a visual representation of the solutions to a polynomial equation. When you graph a polynomial, you can easily identify the x-intercepts, which indicate where the function equals zero. These points, known as real roots, help in solving the polynomial equation.
A few tips for graphing polynomial functions:

• Use a graphing calculator or software for accuracy
• Note the degree of the polynomial, which indicates the maximum number of real roots
• Look for symmetry, as many polynomial functions have symmetrical graphs
• Identify the intercepts, which give critical information about the roots

In the solution, graphing helps us find the approximate real roots at \( x = -2, x = 1, \) and \( x = 2 \).

Polynomial Functions

Polynomial functions are expressions that involve several terms added together, each term consisting of a variable raised to a whole-number exponent. The highest power of the variable in the polynomial is known as the degree, which affects the graph's shape and the number of potential roots.
Key characteristics of polynomial functions include:

• Continuity over the entire real number line, meaning there are no breaks or holes
• Smooth curves, which means no sharp corners or cusps
• The degree determines the number of roots and the end behavior of the graph
• Coefficients and signs affect the graph's orientation and turning points

The given equation in the exercise is a fifth-degree polynomial, signaling that it can have up to five roots.

Real and Complex Solutions

When solving polynomial equations, it's essential to distinguish between real and complex solutions. Real solutions appear as x-intercepts on the graph, while complex solutions do not show up directly and require further algebraic techniques to find.
Understanding real and complex solutions:

• Real solutions are the visible points where the curve crosses the x-axis
• Complex solutions have an imaginary component and come in conjugate pairs
• The number of solutions, both real and complex, must equal the polynomial's degree
• Complex roots do not affect the visible part of the graph but are crucial for a complete solution set

In our problem, after identifying the real solutions and checking them through synthetic division, we must ensure all possible roots (real and complex) are accounted for, totaling five for this degree-five polynomial.