What Is The Root Of The Polynomial Equation $x(x-2)(x+3)=18$?Use A Graphing Calculator And A System Of Equations To Find The Root.A. -3 B. 0 C. 2 D. 3

$$

Introduction

Solving polynomial equations can be a challenging task, especially when dealing with cubic equations. In this article, we will explore two methods to find the root of the polynomial equation $x(x-2)(x+3)=18$. The first method involves using a graphing calculator to visualize the equation and find the root. The second method involves setting up a system of equations to solve for the root.

Method 1: Using a Graphing Calculator

A graphing calculator is a powerful tool that can help us visualize the equation and find the root. To use a graphing calculator, we need to enter the equation and set the calculator to graph the equation.

Step 1: Enter the Equation

Enter the equation $x(x-2)(x+3)=18$ into the graphing calculator. Make sure to use the correct syntax and formatting.

Step 2: Graph the Equation

Graph the equation using the graphing calculator. This will give us a visual representation of the equation and help us identify the root.

Step 3: Find the Root

Use the graphing calculator to find the root of the equation. The root is the x-value where the graph intersects the x-axis.

Method 2: Using a System of Equations

Another method to find the root of the polynomial equation is to set up a system of equations. This method involves setting up two equations and solving for the root.

Step 3.1: Expand the Equation

Expand the equation $x(x-2)(x+3)=18$ to get a cubic equation.

$$x^3 - x^2 - 9x - 18 = 18$$

Step 3.2: Simplify the Equation

Simplify the equation by combining like terms.

$$x^3 - x^2 - 9x - 36 = 0$$

Step 3.3: Set Up a System of Equations

Set up a system of equations by setting the equation equal to zero and factoring.

$$x^3 - x^2 - 9x - 36 = 0$$

$$x(x-2)(x+3) = 0$$

Step 3.4: Solve for the Root

Solve for the root by setting each factor equal to zero and solving for x.

$$x = 0$$

$$x - 2 = 0$$

$$x + 3 = 0$$

Conclusion

In this article, we explored two methods to find the root of the polynomial equation $x(x-2)(x+3)=18$. The first method involved using a graphing calculator to visualize the equation and find the root. The second method involved setting up a system of equations to solve for the root. We found that the root of the equation is x = 2.

Final Answer

The final answer is $\boxed{2}$.

Discussion

The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 2. This can be verified using a graphing calculator or by setting up a system of equations and solving for the root.

Related Topics

• Solving polynomial equations
• Graphing calculators
• Systems of equations

References

• [1] "Solving Polynomial Equations" by Math Open Reference
• [2] "Graphing Calculators" by Texas Instruments
• [3] "Systems of Equations" by Khan Academy
  

Introduction

Solving polynomial equations can be a challenging task, especially when dealing with cubic equations. In this article, we will answer some frequently asked questions about solving polynomial equations.

Q: What is a polynomial equation?

A: A polynomial equation is an equation in which the highest power of the variable (usually x) is a whole number. For example, $x^2 + 3x - 4 = 0$ is a polynomial equation.

Q: How do I solve a polynomial equation?

A: There are several methods to solve a polynomial equation, including factoring, using the quadratic formula, and graphing. The method you choose will depend on the complexity of the equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The formula is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use a graphing calculator to solve a polynomial equation?

A: To use a graphing calculator to solve a polynomial equation, follow these steps:

1. Enter the equation into the calculator.
2. Set the calculator to graph the equation.
3. Use the calculator to find the x-intercepts of the graph.
4. The x-intercepts are the solutions to the equation.

Q: What is the difference between a quadratic equation and a polynomial equation?

A: A quadratic equation is a polynomial equation in which the highest power of the variable is 2. For example, $x^2 + 3x - 4 = 0$ is a quadratic equation. A polynomial equation can have any power of the variable, not just 2.

Q: How do I factor a polynomial equation?

A: Factoring a polynomial equation involves expressing the equation as a product of simpler equations. For example, the equation $x^2 + 5x + 6 = 0$ can be factored as $(x + 3)(x + 2) = 0$.

Q: What is the significance of the x-intercepts of a graph?

A: The x-intercepts of a graph are the points where the graph intersects the x-axis. These points are the solutions to the equation.

Q: Can I use a graphing calculator to solve a cubic equation?

A: Yes, you can use a graphing calculator to solve a cubic equation. However, the calculator may not always be able to find the exact solutions to the equation.

Q: What is the difference between a rational root and an irrational root?

A: A rational root is a root that can be expressed as a fraction, such as 3/4. An irrational root is a root that cannot be expressed as a fraction, such as the square root of 2.

Q: How do I determine if a polynomial equation has a rational root?

A: To determine if a polynomial equation has a rational root, you can use the Rational Root Theorem. This theorem states that if a polynomial equation has a rational root, then that root must be a factor of the constant term.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem is a theorem that states that if a polynomial equation has a rational root, then that root must be a factor of the constant term.

Q: Can I use a graphing calculator to find the rational roots of a polynomial equation?

A: Yes, you can use a graphing calculator to find the rational roots of a polynomial equation. However, the calculator may not always be able to find the exact rational roots.

Q: What is the significance of the degree of a polynomial equation?

A: The degree of a polynomial equation is the highest power of the variable in the equation. For example, the equation $x^3 + 2x^2 - 3x - 1 = 0$ has a degree of 3.

Q: How do I determine the degree of a polynomial equation?

A: To determine the degree of a polynomial equation, you can look at the highest power of the variable in the equation.

Q: Can I use a graphing calculator to determine the degree of a polynomial equation?

A: Yes, you can use a graphing calculator to determine the degree of a polynomial equation. However, the calculator may not always be able to determine the degree of the equation.

Q: What is the difference between a polynomial equation and a rational equation?

A: A polynomial equation is an equation in which the highest power of the variable is a whole number. A rational equation is an equation in which the variable is in the denominator of a fraction.

Q: How do I solve a rational equation?

A: To solve a rational equation, you can cross-multiply and then solve for the variable.

Q: Can I use a graphing calculator to solve a rational equation?

A: Yes, you can use a graphing calculator to solve a rational equation. However, the calculator may not always be able to find the exact solutions to the equation.

Q: What is the significance of the domain of a rational equation?

A: The domain of a rational equation is the set of all possible values of the variable. For example, the equation $1/x = 2$ has a domain of all real numbers except 0.

Q: How do I determine the domain of a rational equation?

A: To determine the domain of a rational equation, you can look at the denominator of the fraction and determine which values of the variable would make the denominator equal to zero.

Q: Can I use a graphing calculator to determine the domain of a rational equation?

A: Yes, you can use a graphing calculator to determine the domain of a rational equation. However, the calculator may not always be able to determine the domain of the equation.

Q: What is the difference between a polynomial equation and a trigonometric equation?

A: A polynomial equation is an equation in which the highest power of the variable is a whole number. A trigonometric equation is an equation that involves trigonometric functions, such as sine and cosine.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you can use trigonometric identities and formulas to simplify the equation and then solve for the variable.

Q: Can I use a graphing calculator to solve a trigonometric equation?

A: Yes, you can use a graphing calculator to solve a trigonometric equation. However, the calculator may not always be able to find the exact solutions to the equation.

Q: What is the significance of the period of a trigonometric equation?

A: The period of a trigonometric equation is the length of time it takes for the graph of the equation to repeat itself.

Q: How do I determine the period of a trigonometric equation?

A: To determine the period of a trigonometric equation, you can use the formula $T = \frac{2\pi}{b}$, where $T$ is the period and $b$ is the coefficient of the variable.

Q: Can I use a graphing calculator to determine the period of a trigonometric equation?

A: Yes, you can use a graphing calculator to determine the period of a trigonometric equation. However, the calculator may not always be able to determine the period of the equation.

Q: What is the difference between a polynomial equation and a system of equations?

A: A polynomial equation is an equation in which the highest power of the variable is a whole number. A system of equations is a set of two or more equations that are solved simultaneously.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use substitution or elimination to solve for the variables.

Q: Can I use a graphing calculator to solve a system of equations?

A: Yes, you can use a graphing calculator to solve a system of equations. However, the calculator may not always be able to find the exact solutions to the equation.

Q: What is the significance of the solution set of a system of equations?

A: The solution set of a system of equations is the set of all possible values of the variables that satisfy the equations.

Q: How do I determine the solution set of a system of equations?

A: To determine the solution set of a system of equations, you can solve the equations simultaneously and then determine the values of the variables that satisfy the equations.

Q: Can I use a graphing calculator to determine the solution set of a system of equations?

A: Yes, you can use a graphing calculator to determine the solution set of a system of equations. However, the calculator may not always be able to determine the solution set of the equation.

Q: What is the difference between a polynomial equation and a matrix equation?

A: A polynomial equation is an equation in which the highest power of the variable is a whole number. A matrix equation is an equation that involves matrices and vectors.

Q: How do I solve a matrix equation?

A: To solve a matrix equation, you can use matrix operations, such as addition and multiplication, to solve for the variables.

Q: Can I use a graphing calculator to solve a matrix equation?

A: Yes, you can use a graphing calculator to solve a matrix equation. However, the calculator may not always be able to find the exact solutions to the equation.

Q: What is the significance of the inverse of a matrix?

A: The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.

Q: How do I determine the inverse of a matrix?

A: