Select The Correct Answer From Each Drop-down Menu.What Is The Factored Form Of This Expression?$8x^3 - 8x^2 - 30x = \square \vee (\square)(\square$\]

Introduction

Factoring expressions is a fundamental concept in algebra that involves breaking down an expression into a product of simpler expressions. It is an essential skill for solving equations, graphing functions, and simplifying complex expressions. In this article, we will explore the factored form of the given expression $8x^3 - 8x^2 - 30x$ and provide a step-by-step guide on how to factor it.

Understanding the Expression

The given expression is $8x^3 - 8x^2 - 30x$. To factor this expression, we need to identify the greatest common factor (GCF) of the terms. The GCF is the largest expression that divides each term without leaving a remainder.

Step 1: Identify the GCF

The GCF of the terms $8x^3$, $8x^2$, and $30x$ is $2x$. We can factor out $2x$ from each term to get:

$$2x(4x^2 - 4x - 15)$$

Step 2: Factor the Quadratic Expression

The quadratic expression $4x^2 - 4x - 15$ can be factored using the quadratic formula or by finding two numbers whose product is $-60$ and whose sum is $-4$. The two numbers are $-10$ and $5$, so we can write:

$$4x^2 - 4x - 15 = (2x + 3)(2x - 5)$$

Step 3: Write the Factored Form

Now that we have factored the quadratic expression, we can write the factored form of the original expression:

$$8x^3 - 8x^2 - 30x = 2x(2x + 3)(2x - 5)$$

Conclusion

Factoring expressions is an essential skill in algebra that involves breaking down an expression into a product of simpler expressions. By identifying the GCF and factoring the quadratic expression, we can write the factored form of the original expression. In this article, we have explored the factored form of the expression $8x^3 - 8x^2 - 30x$ and provided a step-by-step guide on how to factor it.

Tips and Tricks

• Always identify the GCF before factoring the expression.
• Use the quadratic formula or find two numbers whose product is the constant term and whose sum is the coefficient of the linear term to factor the quadratic expression.
• Check your work by multiplying the factors together to ensure that you get the original expression.

Common Mistakes

• Failing to identify the GCF before factoring the expression.
• Not checking your work by multiplying the factors together.
• Factoring the expression incorrectly, resulting in an incorrect factored form.

Real-World Applications

Factoring expressions has numerous real-world applications in fields such as engineering, economics, and computer science. For example, factoring expressions can be used to:

• Solve systems of equations in engineering and physics.
• Model population growth and decline in economics.
• Analyze and optimize complex systems in computer science.

Conclusion

$$

Introduction

Factoring expressions is a fundamental concept in algebra that involves breaking down an expression into a product of simpler expressions. In our previous article, we explored the factored form of the expression $8x^3 - 8x^2 - 30x$ and provided a step-by-step guide on how to factor it. In this article, we will answer some frequently asked questions about factoring expressions and provide additional tips and tricks to help you master this essential skill.

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

A: The GCF of an expression is the largest expression that divides each term without leaving a remainder. It is the product of the common factors of the terms.

Q: How do I identify the GCF of an expression?

A: To identify the GCF of an expression, look for the largest expression that divides each term without leaving a remainder. You can also use the following steps:

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

Q: What is the difference between factoring and simplifying an expression?

A: Factoring an expression involves breaking it down into a product of simpler expressions, while simplifying an expression involves combining like terms to get a simpler expression.

Q: Can I factor an expression with a negative sign?

A: Yes, you can factor an expression with a negative sign. When factoring an expression with a negative sign, you can either factor the expression as is or factor the expression without the negative sign and then multiply the result by -1.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you can use the following 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 by multiplying the binomials together.

Q: Can I factor an expression with a variable in the denominator?

A: No, you cannot factor an expression with a variable in the denominator. When factoring an expression with a variable in the denominator, you need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Q: How do I check my work when factoring an expression?

A: To check your work when factoring an expression, multiply the factors together to ensure that you get the original expression. If the result is not the original expression, you need to re-factor the expression.

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

A: Some common mistakes to avoid when factoring expressions include:

• Failing to identify the GCF before factoring the expression.
• Not checking your work by multiplying the factors together.
• Factoring the expression incorrectly, resulting in an incorrect factored form.

Conclusion

Factoring expressions is a fundamental concept in algebra that involves breaking down an expression into a product of simpler expressions. By identifying the GCF and factoring the quadratic expression, we can write the factored form of the original expression. In this article, we have answered some frequently asked questions about factoring expressions and provided additional tips and tricks to help you master this essential skill.

Tips and Tricks

• Always identify the GCF before factoring the expression.
• Use the quadratic formula or find two numbers whose product is the constant term and whose sum is the coefficient of the linear term to factor the quadratic expression.
• Check your work by multiplying the factors together to ensure that you get the original expression.

Real-World Applications

Factoring expressions has numerous real-world applications in fields such as engineering, economics, and computer science. For example, factoring expressions can be used to:

• Solve systems of equations in engineering and physics.
• Model population growth and decline in economics.
• Analyze and optimize complex systems in computer science.

Conclusion

Factoring expressions is a fundamental concept in algebra that involves breaking down an expression into a product of simpler expressions. By identifying the GCF and factoring the quadratic expression, we can write the factored form of the original expression. In this article, we have answered some frequently asked questions about factoring expressions and provided additional tips and tricks to help you master this essential skill.