Factor $16u + 40v - 32w - 48$. Write Your Answer As A Product With A Whole Number Greater Than 1. □ \square □

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Introduction

In this article, we will focus on factoring the given expression, which is a linear combination of variables u, v, and w. Factoring is a fundamental concept in algebra that involves expressing an expression as a product of simpler expressions. It is an essential skill for solving equations and manipulating expressions in various mathematical contexts.

Understanding the Expression

The given expression is 16u + 40v - 32w - 48. To factor this expression, we need to identify the greatest common factor (GCF) of the coefficients of the variables. The GCF is the largest number that divides all the coefficients without leaving a remainder.

Identifying the Greatest Common Factor (GCF)

The coefficients of the variables are 16, 40, -32, and -48. To find the GCF, we can list the factors of each coefficient:

  • Factors of 16: 1, 2, 4, 8, 16
  • Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
  • Factors of -32: 1, 2, 4, 8, 16, 32
  • Factors of -48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

The greatest common factor of the coefficients is 8.

Factoring the Expression

Now that we have identified the GCF, we can factor the expression by dividing each term by the GCF:

16u + 40v - 32w - 48 = 8(2u + 5v - 4w - 6)

Simplifying the Factored Form

The factored form of the expression is 8(2u + 5v - 4w - 6). We can simplify this expression by combining like terms:

8(2u + 5v - 4w - 6) = 16u + 40v - 32w - 48

Conclusion

In this article, we have factored the given expression, 16u + 40v - 32w - 48, by identifying the greatest common factor (GCF) of the coefficients and dividing each term by the GCF. The factored form of the expression is 8(2u + 5v - 4w - 6). This expression can be simplified by combining like terms.

Example Use Cases

Factoring expressions is an essential skill in various mathematical contexts, including:

  • Solving equations: Factoring expressions can help us solve equations by isolating the variable.
  • Manipulating expressions: Factoring expressions can help us simplify complex expressions and make them easier to work with.
  • Graphing functions: Factoring expressions can help us graph functions by identifying the x-intercepts.

Tips and Tricks

Here are some tips and tricks for factoring expressions:

  • Identify the greatest common factor (GCF) of the coefficients.
  • Divide each term by the GCF.
  • Simplify the factored form by combining like terms.
  • Use factoring to solve equations and manipulate expressions.

Common Mistakes

Here are some common mistakes to avoid when factoring expressions:

  • Failing to identify the greatest common factor (GCF) of the coefficients.
  • Not dividing each term by the GCF.
  • Not simplifying the factored form by combining like terms.

Conclusion

Introduction

In our previous article, we discussed how to factor the expression 16u + 40v - 32w - 48. We identified the greatest common factor (GCF) of the coefficients and divided each term by the GCF to simplify the expression. In this article, we will answer some frequently asked questions about factoring expressions.

Q&A

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

A: The greatest common factor (GCF) of the coefficients is the largest number that divides all the coefficients without leaving a remainder. In the case of the expression 16u + 40v - 32w - 48, the GCF is 8.

Q: How do I identify the greatest common factor (GCF) of the coefficients?

A: To identify the GCF, you can list the factors of each coefficient and find the largest number that is common to all the coefficients. For example, the factors of 16 are 1, 2, 4, 8, and 16. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. The GCF of 16 and 40 is 8.

Q: How do I factor an expression?

A: To factor an expression, you need to identify the greatest common factor (GCF) of the coefficients and divide each term by the GCF. For example, to factor the expression 16u + 40v - 32w - 48, you would divide each term by 8 to get 2u + 5v - 4w - 6.

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. For example, the expression 2u + 5v - 4w - 6 can be simplified by combining like terms to get 2u + 5v - 4w - 6.

Q: How do I know if an expression can be factored?

A: An expression can be factored if it has a greatest common factor (GCF) of the coefficients. If the coefficients do not have a common factor, then the expression cannot be factored.

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 greatest common factor (GCF) of the coefficients.
  • Not dividing each term by the GCF.
  • Not simplifying the factored form by combining like terms.

Q: How do I use factoring to solve equations?

A: Factoring can be used to solve equations by isolating the variable. For example, if you have the equation 2u + 5v - 4w - 6 = 0, you can factor the expression 2u + 5v - 4w - 6 to get 2(u + 2v - 2w - 3) = 0. Then, you can solve for u by isolating it on one side of the equation.

Q: How do I use factoring to manipulate expressions?

A: Factoring can be used to manipulate expressions by simplifying them. For example, if you have the expression 2u + 5v - 4w - 6, you can factor it to get 2(u + 2v - 2w - 3). Then, you can simplify the expression by combining like terms to get 2u + 5v - 4w - 6.

Conclusion

In conclusion, factoring expressions is an essential skill in algebra that involves expressing an expression as a product of simpler expressions. By identifying the greatest common factor (GCF) of the coefficients and dividing each term by the GCF, we can factor expressions and simplify complex expressions. With practice and patience, you can master the art of factoring expressions and become proficient in solving equations and manipulating expressions.

Example Use Cases

Factoring expressions is an essential skill in various mathematical contexts, including:

  • Solving equations: Factoring expressions can help us solve equations by isolating the variable.
  • Manipulating expressions: Factoring expressions can help us simplify complex expressions and make them easier to work with.
  • Graphing functions: Factoring expressions can help us graph functions by identifying the x-intercepts.

Tips and Tricks

Here are some tips and tricks for factoring expressions:

  • Identify the greatest common factor (GCF) of the coefficients.
  • Divide each term by the GCF.
  • Simplify the factored form by combining like terms.
  • Use factoring to solve equations and manipulate expressions.

Common Mistakes

Here are some common mistakes to avoid when factoring expressions:

  • Failing to identify the greatest common factor (GCF) of the coefficients.
  • Not dividing each term by the GCF.
  • Not simplifying the factored form by combining like terms.