Problem 122 - College Algebra

Question

Use a graphing utility and the graph's \(x\) -intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$\begin{aligned} &x^{3}+3 x^{2}-x-3-0\\\ &[-6,6,1] \text { by }[-6,6,1] \end{aligned}$$

Step-by-Step Solution

Answer

To determine the roots of a polynomial, graph the function, identify the x-intercepts (which correspond to the roots), and then verify the roots by direct substitution into the function.

Step 1: Plot the Graph

Use a graphing utility to graph the function \(f(x) = x^3 + 3x^2 - x - 3\) in the specified viewing rectangle: x-coordinate varies from -6 to 6, and y-coordinate varies from -6 to 6.

Step 2: Find the x-intercepts

The x-intercepts of the function correspond to the roots of the equation \(x^3 + 3x^2 - x - 3 = 0\). These are the x-values where the graph crosses the x-axis. Note down these values.

Step 3: Verify the Roots

Substitute the x-values obtained from the graph one by one back into the equation to verify if they are accurate. They should satisfy the equation \(x^3 + 3x^2 - x - 3 = 0\).

Key Concepts

Understanding X-Intercepts

The concept of x-intercepts is essential when dealing with graphs of functions. An x-intercept is a point where the graph of a function crosses the x-axis. At these points, the y-value is always zero. Knowing x-intercepts helps us in solving equations graphically.

For example, in the function given by \(f(x) = x^3 + 3x^2 - x - 3\), our task is to find out the x-values for which \(f(x) = 0\). These are precisely the points where the graph intersects the x-axis.

• X-intercepts can be found visually using a graphing calculator or software.
• These points give us important clues about the roots of the corresponding equation.

When using a graphing calculator or any utility, the x-values where the curve cuts the x-axis are the x-intercepts. By knowing these, we get a graphical solution to the equation’s roots.

Exploring Roots of an Equation

Roots of an equation are the solutions to the equation set to zero. In simpler terms, if you have a function \(f(x)\) and you solve \(f(x) = 0\), the solutions are your roots. These points are also known as "zeros" of the function.

For our polynomial function \(x^3 + 3x^2 - x - 3\), the roots are the values of \(x\) that make the equation zero. Determining these roots gives us important algebraic information:

• By finding the roots, you can determine where the graph intersects the x-axis.
• Each root corresponds to an x-intercept on the graph.
• Verifying these roots can be done by substituting them back into the original equation and ensuring the result equals zero.

When using a graphing tool, marking the roots effectively helps identify the intersection points, confirming them as x-intercepts.

Introduction to Polynomial Functions

Polynomial functions are algebraic expressions made up of terms in the form of \(a_nx^n\), where each term is a power of \(x\) multiplied by a coefficient. The highest power of \(x\) determines the function's degree.
Our specific function, \(x^3 + 3x^2 - x - 3\), is a cubic polynomial because the highest power of \(x\) is 3.
Polynomials have a variety of characteristics:

• The degree of a polynomial tells how many roots or x-intercepts it can have. A cubic polynomial, like our function, may have up to three real roots.
• The coefficient of the highest degree term, which is 1 in our example, is also known as the leading coefficient.
• Polynomials are continuous and smooth, meaning they have no breaks or sharp corners.

Graphing these functions, especially with a graphing calculator, aids in quickly identifying important features like roots and intercepts.