Factor Completely.12u3v 15u3–60u2v–75u2

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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: 12u3v 15u3–60u2v–75u2. We will break down the expression into its prime factors and then simplify it to its completely factored form.

Step 1: Factor out the Greatest Common Factor (GCF)

The first step in factoring the given expression is to identify the greatest common factor (GCF) of the three terms. The GCF is the largest expression that divides each term without leaving a remainder.

12u3v = 3u3v
15u3 = 3u3
60u2v = 12u2v
75u2 = 15u2

The GCF of the three terms is 3u2v.

Step 2: Factor out the GCF from each term

Now that we have identified the GCF, we can factor it out from each term.

12u3v = 3u2v \* 4u
15u3 = 3u2v \* 5u
60u2v = 3u2v \* 20
75u2 = 3u2v \* 25

Step 3: Combine like terms

Now that we have factored out the GCF from each term, we can combine like terms.

3u2v \* 4u + 3u2v \* 5u + 3u2v \* 20 - 3u2v \* 25

We can combine the like terms by adding or subtracting the coefficients.

3u2v \* (4u + 5u + 20 - 25)

Step 4: Simplify the expression

Now that we have combined like terms, we can simplify the expression.

3u2v \* (9u - 5)

Conclusion

In this article, we have factored the given expression: 12u3v 15u3–60u2v–75u2. We have broken down the expression into its prime factors and then simplified it to its completely factored form. The final factored form of the expression is 3u2v * (9u - 5).

Tips and Tricks

  • When factoring an expression, it is essential to identify the greatest common factor (GCF) of the terms.
  • Once you have identified the GCF, you can factor it out from each term.
  • Combine like terms to simplify the expression.
  • Use the distributive property to expand the expression and check your work.

Common Mistakes to Avoid

  • Not identifying the greatest common factor (GCF) of the terms.
  • Not factoring out the GCF from each term.
  • Not combining like terms to simplify the expression.
  • Not using the distributive property to expand the expression and check your work.

Real-World Applications

Factoring is a fundamental concept in algebra that has numerous real-world applications. Some of the real-world applications of factoring include:

  • Science: Factoring is used in science to solve equations and model real-world phenomena.
  • Engineering: Factoring is used in engineering to design and optimize systems.
  • Economics: Factoring is used in economics to model economic systems and make predictions.
  • Computer Science: Factoring is used in computer science to develop algorithms and solve problems.

Conclusion

Introduction

In our previous article, we factored the given expression: 12u3v 15u3–60u2v–75u2. We broke down the expression into its prime factors and then simplified it to its completely factored form. In this article, we will answer some frequently asked questions (FAQs) related to factoring.

Q: What is factoring?

A: Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions.

Q: Why is factoring important?

A: Factoring is important because it helps us to simplify complex expressions and solve equations. It is also used in various real-world applications such as science, engineering, economics, and computer science.

Q: How do I factor an expression?

A: To factor an expression, you need to identify the greatest common factor (GCF) of the terms. Once you have identified the GCF, you can factor it out from each term. Then, combine like terms to simplify the expression.

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

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

Q: How do I identify the GCF?

A: To identify the GCF, you need to list the factors of each term and find the common factors. The GCF is the product of the common factors.

Q: Can I factor an expression with variables?

A: Yes, you can factor an expression with variables. The process of factoring is the same as factoring an expression with constants.

Q: Can I factor an expression with fractions?

A: Yes, you can factor an expression with fractions. The process of factoring is the same as factoring an expression with constants.

Q: How do I check my work when factoring?

A: To check your work when factoring, you need to use the distributive property to expand the expression and check if it is equal to the original expression.

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

A: Some common mistakes to avoid when factoring include:

  • Not identifying the greatest common factor (GCF) of the terms.
  • Not factoring out the GCF from each term.
  • Not combining like terms to simplify the expression.
  • Not using the distributive property to expand the expression and check your work.

Q: Can I use a calculator to factor an expression?

A: Yes, you can use a calculator to factor an expression. However, it is always a good idea to check your work by hand to ensure that the calculator is giving you the correct answer.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to identify the greatest common factor (GCF) of the terms. Once you have identified the GCF, you can factor it out from each term. Then, use the quadratic formula to factor the expression.

Q: Can I factor a polynomial expression?

A: Yes, you can factor a polynomial expression. The process of factoring is the same as factoring an expression with constants.

Conclusion

In conclusion, factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. In this article, we have answered some frequently asked questions (FAQs) related to factoring. We hope that this article has been helpful in understanding the concept of factoring and how to apply it in various real-world applications.

Tips and Tricks

  • When factoring an expression, it is essential to identify the greatest common factor (GCF) of the terms.
  • Once you have identified the GCF, you can factor it out from each term.
  • Combine like terms to simplify the expression.
  • Use the distributive property to expand the expression and check your work.
  • Check your work by hand to ensure that the calculator is giving you the correct answer.

Common Mistakes to Avoid

  • Not identifying the greatest common factor (GCF) of the terms.
  • Not factoring out the GCF from each term.
  • Not combining like terms to simplify the expression.
  • Not using the distributive property to expand the expression and check your work.
  • Not checking your work by hand to ensure that the calculator is giving you the correct answer.