Factor The Expression Completely.\[$-18 + 42x\$\]

by ADMIN 50 views

Understanding the Problem

Factoring an expression is a fundamental concept in algebra that involves breaking down an expression into simpler components. In this article, we will focus on factoring the expression completely, which means expressing the given expression as a product of its prime factors.

The Given Expression

The given expression is −18+42x{-18 + 42x}. Our goal is to factor this expression completely.

Step 1: Identify the Greatest Common Factor (GCF)

To factor the expression completely, we need to identify the greatest common factor (GCF) of the two terms. The GCF is the largest factor that divides both terms without leaving a remainder.

In this case, the two terms are -18 and 42x. We can start by finding the factors of each term.

  • Factors of -18: -1, 1, 2, 3, 6, 9, 18
  • Factors of 42x: 1, 2, 3, 6, 7, 14, 21, 42, x, 2x, 3x, 6x, 7x, 14x, 21x, 42x

Now, we need to find the greatest common factor of these two sets of factors.

The greatest common factor of -18 and 42x is 6.

Step 2: Factor Out the GCF

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

−18+42x=6(−3+7x){-18 + 42x = 6(-3 + 7x)}

In this step, we have factored out the GCF (6) from each term.

Step 3: Check for Further Factoring

Now that we have factored out the GCF, we need to check if there are any further factors that can be factored out.

In this case, we can see that the expression −3+7x{-3 + 7x} cannot be factored further.

Therefore, the final factored form of the expression is 6(−3+7x){6(-3 + 7x)}.

Conclusion

In this article, we have learned how to factor the expression completely. We started by identifying the GCF of the two terms, factored it out, and then checked for further factoring.

The final factored form of the expression is 6(−3+7x){6(-3 + 7x)}.

Example Problems

Here are some example problems that you can try to practice factoring expressions completely.

Example 1

Factor the expression completely: 24+40x{24 + 40x}

Solution

To factor the expression completely, we need to identify the GCF of the two terms.

The GCF of 24 and 40x is 8.

Now, we can factor it out of each term.

24+40x=8(3+5x){24 + 40x = 8(3 + 5x)}

Example 2

Factor the expression completely: −36+48x{-36 + 48x}

Solution

To factor the expression completely, we need to identify the GCF of the two terms.

The GCF of -36 and 48x is 12.

Now, we can factor it out of each term.

−36+48x=12(−3+4x){-36 + 48x = 12(-3 + 4x)}

Example 3

Factor the expression completely: 20+30x{20 + 30x}

Solution

To factor the expression completely, we need to identify the GCF of the two terms.

The GCF of 20 and 30x is 10.

Now, we can factor it out of each term.

20+30x=10(2+3x){20 + 30x = 10(2 + 3x)}

Tips and Tricks

Here are some tips and tricks that you can use to help you factor expressions completely.

  • Always start by identifying the GCF of the two terms.
  • Factor the GCF out of each term.
  • Check for further factoring by looking for common factors in the remaining terms.
  • Use the distributive property to expand the expression and check for further factoring.

By following these tips and tricks, you can become proficient in factoring expressions completely.

Common Mistakes to Avoid

Here are some common mistakes to avoid when factoring expressions completely.

  • Not identifying the GCF of the two terms.
  • Not factoring the GCF out of each term.
  • Not checking for further factoring.
  • Not using the distributive property to expand the expression and check for further factoring.

By avoiding these common mistakes, you can ensure that you are factoring expressions completely correctly.

Conclusion

In this article, we have learned how to factor the expression completely. We started by identifying the GCF of the two terms, factored it out, and then checked for further factoring.

The final factored form of the expression is 6(−3+7x){6(-3 + 7x)}.

We have also provided example problems and tips and tricks to help you practice factoring expressions completely.

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about factoring expressions completely.

Q: What is factoring an expression?

A: Factoring an expression is the process of breaking down an expression into simpler components. It involves expressing the given expression as a product of its prime factors.

Q: Why is factoring important?

A: Factoring is an important concept in algebra because it helps us to simplify complex expressions and solve equations. It is also a crucial step in solving quadratic equations and other types of equations.

Q: How do I factor an expression?

A: To factor an expression, you need to follow these steps:

  1. Identify the greatest common factor (GCF) of the two terms.
  2. Factor the GCF out of each term.
  3. Check for further factoring by looking for common factors in the remaining terms.
  4. Use the distributive property to expand the expression and check for further factoring.

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

A: The greatest common factor (GCF) is the largest factor that divides both terms without leaving a remainder. It is the product of the common factors of the two terms.

Q: How do I find the GCF of two terms?

A: To find the GCF of two terms, you need to list the factors of each term and then identify the greatest common factor.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a single term can be distributed to multiple terms. It is used to expand expressions and check for further factoring.

Q: How do I use the distributive property to expand an expression?

A: To use the distributive property to expand an expression, you need to multiply each term in the expression by the factor that is being distributed.

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 the two terms.
  • Not factoring the GCF out of each term.
  • Not checking for further factoring.
  • Not using the distributive property to expand the expression and check for further factoring.

Q: How can I practice factoring expressions?

A: You can practice factoring expressions by working on example problems and exercises. You can also use online resources and practice tests to help you improve your skills.

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

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

  • Simplifying complex expressions in physics and engineering.
  • Solving quadratic equations in finance and economics.
  • Analyzing data in statistics and data analysis.

Conclusion

In this article, we have answered some of the most frequently asked questions about factoring expressions completely. We have covered topics such as the greatest common factor, the distributive property, and common mistakes to avoid.

By following the steps and tips outlined in this article, you can become proficient in factoring expressions completely. Remember to practice regularly and use online resources to help you improve your skills.

Additional Resources

Here are some additional resources that you can use to help you learn more about factoring expressions completely.

  • Online tutorials and videos
  • Practice tests and exercises
  • Online communities and forums
  • Textbooks and study guides

By using these resources, you can improve your skills and become proficient in factoring expressions completely.