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

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Introduction

Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. In this article, we will focus on factoring the given expression 8x3−8x2−30x8x^3 - 8x^2 - 30x. We will break down the process into manageable steps and provide a clear explanation of each step.

Step 1: Identify the Greatest Common Factor (GCF)

The first step in factoring the given expression is to identify the greatest common factor (GCF) of the terms. The GCF is the largest expression that divides each term without leaving a remainder. In this case, the GCF of the terms 8x38x^3, −8x2-8x^2, and −30x-30x is 2x2x.

Factoring Out the GCF

To factor out the GCF, we divide each term by the GCF and write the result as a product of the GCF and the remaining expression.

8x^3 - 8x^2 - 30x = 2x(4x^2 - 4x - 15)

Step 2: Factor the Quadratic Expression

The next step is to factor the quadratic expression 4x2−4x−154x^2 - 4x - 15. To do this, we need to find two numbers whose product is −60-60 (the product of the coefficient of x2x^2 and the constant term) and whose sum is −4-4 (the coefficient of xx).

Finding the Factors

After some trial and error, we find that the two numbers are −10-10 and 66. Therefore, we can write the quadratic expression as:

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 as:

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

Conclusion

In this article, we have factored the given expression 8x3−8x2−30x8x^3 - 8x^2 - 30x using the greatest common factor (GCF) and factoring quadratic expressions. We have broken down the process into manageable steps and provided a clear explanation of each step. By following these steps, you should be able to factor the given expression and write it in its factored form.

Final Answer

The factored form of the expression 8x3−8x2−30x8x^3 - 8x^2 - 30x is:

2x×(2x+3)×(2x−5){ 2x \times (2x + 3) \times (2x - 5) }

Discussion

Factoring is an essential concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. In this article, we have focused on factoring the given expression 8x3−8x2−30x8x^3 - 8x^2 - 30x using the greatest common factor (GCF) and factoring quadratic expressions. We have broken down the process into manageable steps and provided a clear explanation of each step.

Related Topics

  • Greatest Common Factor (GCF)
  • Factoring Quadratic Expressions
  • Algebraic Expressions
  • Factoring Techniques

References

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Introduction

Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. In our previous article, we focused on factoring the given expression 8x3−8x2−30x8x^3 - 8x^2 - 30x using the greatest common factor (GCF) and factoring quadratic expressions. In this article, we will provide a Q&A guide to help you understand the concept of factoring and how to apply it to different types of expressions.

Q&A

Q1: What is factoring?

A: Factoring is the process of expressing an algebraic expression as a product of simpler expressions.

Q2: Why is factoring important?

A: Factoring is important because it allows us to simplify complex expressions and solve equations more easily.

Q3: What are the different types of factoring?

A: There are several types of factoring, including:

  • Greatest Common Factor (GCF) factoring
  • Factoring quadratic expressions
  • Factoring polynomial expressions
  • Factoring rational expressions

Q4: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the product of the coefficient of x2x^2 and the constant term, and whose sum is the coefficient of xx.

Q5: What is the greatest common factor (GCF)?

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

Q6: How do I find the GCF of an expression?

A: To find the GCF of an expression, you need to list all the factors of each term and find the greatest common factor.

Q7: Can you give an example of factoring a polynomial expression?

A: Yes, here is an example of factoring a polynomial expression:

2x3+6x2+4x+12=2(x3+3x2+2x+6){ 2x^3 + 6x^2 + 4x + 12 = 2(x^3 + 3x^2 + 2x + 6) }

Q8: How do I factor a rational expression?

A: To factor a rational expression, you need to factor the numerator and denominator separately and then simplify the expression.

Q9: 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 (GCF)
  • Not factoring quadratic expressions correctly
  • Not simplifying the expression after factoring

Q10: How can I practice factoring?

A: You can practice factoring by working through examples and exercises in your textbook or online resources.

Conclusion

In this article, we have provided a Q&A guide to help you understand the concept of factoring and how to apply it to different types of expressions. We have covered topics such as the greatest common factor (GCF), factoring quadratic expressions, and factoring polynomial and rational expressions. By practicing factoring, you can become more confident and proficient in solving algebraic equations.

Final Answer

The factored form of the expression 8x3−8x2−30x8x^3 - 8x^2 - 30x is:

2x×(2x+3)×(2x−5){ 2x \times (2x + 3) \times (2x - 5) }

Discussion

Factoring is an essential concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. In this article, we have focused on providing a Q&A guide to help you understand the concept of factoring and how to apply it to different types of expressions.

Related Topics

  • Greatest Common Factor (GCF)
  • Factoring Quadratic Expressions
  • Factoring Polynomial Expressions
  • Factoring Rational Expressions

References