Factor The Expression:$8x^5 + 4x^2 - 12$A. $8(x^5 + 4x^2 - 12$\] B. $4(2x^5 + X^2 - 3$\] C. $4(2x^5 + 4x^2 - 12$\] D. $4x^2(2x^3 + X - 3$\]

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Introduction

Factoring an expression is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. In this article, we will focus on factoring the expression 8x5+4x2−128x^5 + 4x^2 - 12. Factoring expressions is an essential skill in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics.

What is Factoring?

Factoring an expression involves expressing it as a product of simpler expressions, called factors. These factors can be numbers, variables, or a combination of both. Factoring an expression can help us simplify it, make it easier to solve, and even identify its roots.

The Expression to be Factored

The expression we need to factor is 8x5+4x2−128x^5 + 4x^2 - 12. This expression consists of three terms: 8x58x^5, 4x24x^2, and −12-12. To factor this expression, we need to find a common factor that can be factored out from all three terms.

Step 1: Identify the Greatest Common Factor (GCF)

The first step in factoring an expression is to identify the greatest common factor (GCF) of all the terms. The GCF is the largest factor that can be divided evenly into all the terms. In this case, the GCF of 8x58x^5, 4x24x^2, and −12-12 is 44.

Step 2: Factor Out the GCF

Once we have identified the GCF, we can factor it out from each term. To do this, we divide each term by the GCF and write the result as a product of the GCF and the remaining factor.

8x^5 ÷ 4 = 2x^5
4x^2 ÷ 4 = x^2
-12 ÷ 4 = -3

Step 3: Write the Factored Form

Now that we have factored out the GCF, we can write the factored form of the expression. The factored form is obtained by multiplying the GCF by the remaining factors.

4(2x^5 + x^2 - 3)

Conclusion

In this article, we have factored the expression 8x5+4x2−128x^5 + 4x^2 - 12. We identified the GCF of all the terms, factored it out, and wrote the factored form of the expression. Factoring expressions is an essential skill in mathematics, and it has numerous applications in various fields. By following the steps outlined in this article, you can factor expressions with ease and simplify complex mathematical expressions.

Answer

The correct answer is:

B. 4(2x5+x2−3)4(2x^5 + x^2 - 3)

Additional Examples

Here are some additional examples of factoring expressions:

  • 6x3+9x2−126x^3 + 9x^2 - 12
  • 3x2+6x−93x^2 + 6x - 9
  • 2x4+4x2−122x^4 + 4x^2 - 12

Tips and Tricks

Here are some tips and tricks to help you factor expressions:

  • Identify the GCF of all the terms.
  • Factor out the GCF from each term.
  • Write the factored form of the expression.
  • Use the distributive property to check your answer.

Common Mistakes

Here are some common mistakes to avoid when factoring expressions:

  • Not identifying the GCF of all the terms.
  • Not factoring out the GCF from each term.
  • Writing the factored form incorrectly.

Conclusion

Introduction

Factoring expressions is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. In our previous article, we discussed the steps involved in factoring the expression 8x5+4x2−128x^5 + 4x^2 - 12. In this article, we will answer some frequently asked questions about factoring expressions.

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

A: The greatest common factor (GCF) is the largest factor that can be divided evenly into all the terms of an expression.

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

A: To identify the GCF of an expression, you need to find the largest factor that can be divided evenly into all the terms. You can do this by listing the factors of each term and finding the greatest common factor.

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

A: Factoring an expression involves expressing it as a product of simpler expressions, while simplifying an expression involves combining like terms to make it easier to work with.

Q: Can I factor an expression that has no common factors?

A: Yes, you can factor an expression that has no common factors. In this case, you can use other factoring techniques, such as factoring by grouping or using the distributive property.

Q: How do I factor an expression with variables?

A: To factor an expression with variables, you need to identify the GCF of the variables and factor it out. You can also use other factoring techniques, such as factoring by grouping or using the distributive property.

Q: Can I factor an expression with negative coefficients?

A: Yes, you can factor an expression with negative coefficients. In this case, you can use the distributive property to factor out the negative coefficient.

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

A: To check your answer when factoring an expression, you can use the distributive property to multiply the factors and see if you get the original expression.

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

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

  • Not identifying the GCF of all the terms.
  • Not factoring out the GCF from each term.
  • Writing the factored form incorrectly.

Q: How can I practice factoring expressions?

A: You can practice factoring expressions by working through examples and exercises. You can also use online resources, such as factoring calculators or worksheets, to help you practice.

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

A: Factoring expressions has numerous real-world applications, including:

  • Physics: Factoring expressions is used to solve problems involving motion, energy, and momentum.
  • Engineering: Factoring expressions is used to design and optimize systems, such as bridges and buildings.
  • Economics: Factoring expressions is used to model and analyze economic systems.

Conclusion

Factoring expressions is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. By following the steps outlined in this article, you can factor expressions with ease and simplify complex mathematical expressions. Remember to identify the GCF of all the terms, factor it out, and write the factored form of the expression. With practice and patience, you can become proficient in factoring expressions and tackle complex mathematical problems with confidence.

Additional Resources

Here are some additional resources to help you practice factoring expressions:

  • Factoring calculators: Online calculators that can help you factor expressions.
  • Factoring worksheets: Printable worksheets that provide examples and exercises to help you practice factoring expressions.
  • Factoring videos: Online videos that provide step-by-step instructions and examples of factoring expressions.

Common Factoring Techniques

Here are some common factoring techniques:

  • Factoring by grouping: This involves grouping terms together and factoring out common factors.
  • Factoring by difference of squares: This involves factoring expressions that can be written as the difference of squares.
  • Factoring by sum and difference: This involves factoring expressions that can be written as the sum or difference of two terms.

Conclusion

Factoring expressions is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. By following the steps outlined in this article, you can factor expressions with ease and simplify complex mathematical expressions. Remember to identify the GCF of all the terms, factor it out, and write the factored form of the expression. With practice and patience, you can become proficient in factoring expressions and tackle complex mathematical problems with confidence.