Solve The Equation By Factoring: 2 X 3 − 2 X 2 − 40 X = 0 2x^3 - 2x^2 - 40x = 0 2 X 3 − 2 X 2 − 40 X = 0

Introduction

In this article, we will focus on solving a cubic equation by factoring. The given equation is $2x^3 - 2x^2 - 40x = 0$. Factoring is a powerful technique used to simplify and solve polynomial equations. It involves expressing the equation as a product of simpler expressions, called factors. In this case, we will use factoring to find the solutions of the given cubic equation.

Understanding the Equation

Before we proceed with factoring, let's analyze the given equation. The equation is a cubic equation, meaning it has a degree of 3. The general form of a cubic equation is $ax^3 + bx^2 + cx + d = 0$, where $a$, $b$, $c$, and $d$ are constants. In this case, the equation is $2x^3 - 2x^2 - 40x = 0$. We can see that the coefficient of $x^3$ is 2, the coefficient of $x^2$ is -2, and the coefficient of $x$ is -40.

Factoring Out the Greatest Common Factor

The first step in factoring is to identify the greatest common factor (GCF) of the terms. In this case, the GCF is 2x. We can factor out 2x from each term:

$$2x^3 - 2x^2 - 40x = 2x(x^2 - x - 20)$$

Factoring the Quadratic Expression

Now we have a quadratic expression inside the parentheses: $x^2 - x - 20$. We can factor this expression by finding two numbers whose product is -20 and whose sum is -1. The numbers are -5 and 4, so we can write:

$$x^2 - x - 20 = (x - 5)(x + 4)$$

Combining the Factors

Now we can combine the factors we have found:

$$2x(x^2 - x - 20) = 2x(x - 5)(x + 4)$$

Solving for x

To find the solutions of the equation, we set each factor equal to zero and solve for x:

$$2x = 0 \Rightarrow x = 0$$

$$(x - 5) = 0 \Rightarrow x = 5$$

$$(x + 4) = 0 \Rightarrow x = -4$$

Conclusion

In this article, we have solved the cubic equation $2x^3 - 2x^2 - 40x = 0$ by factoring. We first factored out the greatest common factor, then factored the quadratic expression inside the parentheses. Finally, we combined the factors and solved for x. The solutions of the equation are x = 0, x = 5, and x = -4.

Tips and Tricks

• When factoring, always look for the greatest common factor first.
• When factoring a quadratic expression, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
• When solving for x, make sure to set each factor equal to zero and solve for x.

Real-World Applications

Factoring is a powerful technique used in many real-world applications, including:

• Science: Factoring is used to solve equations that model real-world phenomena, such as the motion of objects or the growth of populations.
• Engineering: Factoring is used to design and optimize systems, such as electrical circuits or mechanical systems.
• Economics: Factoring is used to model and analyze economic systems, such as supply and demand curves.

Common Mistakes

• Not factoring out the greatest common factor: This can lead to unnecessary complexity and make it harder to solve the equation.
• Not factoring the quadratic expression: This can lead to missing solutions or incorrect solutions.
• Not checking for extraneous solutions: This can lead to incorrect solutions or solutions that do not satisfy the original equation.

Conclusion

Introduction

In our previous article, we solved the cubic equation $2x^3 - 2x^2 - 40x = 0$ by factoring. In this article, we will provide a Q&A guide to help you understand the concept of factoring and how to apply it to solve polynomial equations.

Q: What is factoring?

A: Factoring is a technique used to simplify and solve polynomial equations by expressing them as a product of simpler expressions, called factors.

Q: Why is factoring important?

A: Factoring is important because it allows us to simplify complex equations and find their solutions more easily. It is also a powerful tool for solving polynomial equations, which are used to model real-world phenomena.

Q: How do I factor an equation?

A: To factor an equation, follow these steps:

1. Identify the greatest common factor (GCF) of the terms.
2. Factor out the GCF from each term.
3. Look for a quadratic expression inside the parentheses.
4. Factor the quadratic expression by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
5. Combine the factors to find the solutions of the equation.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest expression that divides each term of the equation without leaving a remainder.

Q: How do I find the GCF?

A: To find the GCF, look for the largest expression that divides each term of the equation. You can use the following steps:

1. List the factors of each term.
2. Identify the common factors.
3. Multiply the common factors to find the GCF.

Q: What is a quadratic expression?

A: A quadratic expression is a polynomial expression of degree 2, which means it has a squared variable.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, follow these steps:

1. Look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
2. Write the quadratic expression as a product of two binomials.
3. Simplify the expression to find the factored form.

Q: What are some common mistakes to avoid when factoring?

A: Some common mistakes to avoid when factoring include:

• Not factoring out the greatest common factor.
• Not factoring the quadratic expression.
• Not checking for extraneous solutions.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, plug the solutions back into the original equation and check if they satisfy the equation.

Q: What are some real-world applications of factoring?

A: Factoring has many real-world applications, including:

• Science: Factoring is used to solve equations that model real-world phenomena, such as the motion of objects or the growth of populations.
• Engineering: Factoring is used to design and optimize systems, such as electrical circuits or mechanical systems.
• Economics: Factoring is used to model and analyze economic systems, such as supply and demand curves.

Conclusion

In conclusion, factoring is a powerful technique used to solve polynomial equations. By following the steps outlined in this article, you can solve cubic equations and other polynomial equations. Remember to always look for the greatest common factor, factor the quadratic expression, and combine the factors to find the solutions of the equation.