Now, Factor The Expression.$\[ \begin{aligned} 3b + 4 + 10 + 6b - 2 & = 9b + 12 \\ & = 3(\square B + \square ?) \end{aligned} \\]

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. In this article, we will focus on factoring expressions, which is a fundamental concept in algebra. We will use a specific example to demonstrate how to factor an expression and provide a step-by-step guide on how to simplify algebraic expressions.

The Expression to Factor

The given expression is:

$$3b + 4 + 10 + 6b - 2 = 9b + 12$$

Our goal is to factor this expression into the form $3(\square b + \square ?)$.

Step 1: Combine Like Terms

The first step in simplifying the expression is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two like terms: $3b$ and $6b$. We can combine these terms by adding their coefficients:

$$3b + 6b = 9b$$

Now, let's rewrite the expression with the combined like terms:

$$9b + 4 + 10 - 2 = 9b + 12$$

Step 2: Simplify the Constant Terms

Next, we need to simplify the constant terms. We can do this by combining the constants:

$$4 + 10 - 2 = 12$$

Now, let's rewrite the expression with the simplified constant terms:

$$9b + 12$$

Step 3: Factor Out the Greatest Common Factor (GCF)

The next step is to factor out the greatest common factor (GCF) of the terms. In this case, the GCF of $9b$ and $12$ is $3$. We can factor out the GCF by dividing each term by $3$:

$$9b = 3 \times 3b$$

$$12 = 3 \times 4$$

Now, let's rewrite the expression with the factored terms:

$$3 \times 3b + 3 \times 4$$

Step 4: Write the Factored Form

Finally, we can write the factored form of the expression:

$$3(3b + 4)$$

Conclusion

In this article, we demonstrated how to factor an expression using a step-by-step guide. We combined like terms, simplified constant terms, and factored out the greatest common factor (GCF) to arrive at the factored form of the expression. By following these steps, you can simplify algebraic expressions and solve equations and inequalities.

Tips and Tricks

• When combining like terms, make sure to add the coefficients of the like terms.
• When simplifying constant terms, make sure to combine the constants.
• When factoring out the GCF, make sure to divide each term by the GCF.
• When writing the factored form, make sure to use parentheses to group the terms.

Common Mistakes to Avoid

• Not combining like terms correctly.
• Not simplifying constant terms correctly.
• Not factoring out the GCF correctly.
• Not writing the factored form correctly.

Real-World Applications

Factoring expressions has many real-world applications, including:

• Solving equations and inequalities in physics and engineering.
• Modeling population growth and decline in biology.
• Analyzing data in statistics and data science.
• Solving optimization problems in economics and finance.

Practice Problems

Here are some practice problems to help you reinforce your understanding of factoring expressions:

1. Factor the expression: $2x + 5 + 3x - 2$
2. Factor the expression: $4y - 3 + 2y + 1$
3. Factor the expression: $6z + 2 + 3z - 4$

Answer Key

1. $5x + 3$
2. $6y - 2$
3. $9z - 2$

Conclusion

Q: What is factoring an expression?

A: Factoring an expression is the process of expressing it as a product of simpler expressions, called factors. This is done by identifying the greatest common factor (GCF) of the terms and factoring it out.

Q: Why is factoring important?

A: Factoring is important because it helps us simplify algebraic expressions, which is a crucial skill in solving equations and inequalities. By factoring expressions, we can identify the underlying structure of the expression and make it easier to work with.

Q: How do I factor an expression?

A: To factor an expression, follow these steps:

1. Combine like terms.
2. Simplify constant terms.
3. Factor out the greatest common factor (GCF).
4. Write the factored form.

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

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

Q: How do I find the GCF?

A: To find the GCF, look for the largest expression that divides each term of the expression. You can use the following methods:

• List the factors of each term and find the common factors.
• Use the prime factorization method to find the GCF.

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

A: Some common mistakes to avoid when factoring include:

• Not combining like terms correctly.
• Not simplifying constant terms correctly.
• Not factoring out the GCF correctly.
• Not writing the factored form correctly.

Q: How do I check if my factored form is correct?

A: To check if your factored form is correct, multiply the factors together and simplify the expression. If the result is the original expression, then your factored form is correct.

Q: Can I factor expressions with variables?

A: Yes, you can factor expressions with variables. However, you need to follow the same steps as factoring expressions with constants.

Q: Can I factor expressions with fractions?

A: Yes, you can factor expressions with fractions. However, you need to follow the same steps as factoring expressions with constants, and also consider the fraction as a single term.

Q: How do I factor expressions with exponents?

A: To factor expressions with exponents, follow the same steps as factoring expressions with constants, and also consider the exponent as a single term.

Q: Can I factor expressions with negative numbers?

A: Yes, you can factor expressions with negative numbers. However, you need to follow the same steps as factoring expressions with constants, and also consider the negative sign as a single term.

Q: How do I factor expressions with decimals?

A: To factor expressions with decimals, follow the same steps as factoring expressions with constants, and also consider the decimal as a single term.

Q: Can I factor expressions with mixed numbers?

A: Yes, you can factor expressions with mixed numbers. However, you need to follow the same steps as factoring expressions with constants, and also consider the mixed number as a single term.

Q: How do I factor expressions with imaginary numbers?

A: To factor expressions with imaginary numbers, follow the same steps as factoring expressions with constants, and also consider the imaginary number as a single term.

Conclusion

In conclusion, factoring expressions is a crucial skill in algebra that helps us simplify algebraic expressions and solve equations and inequalities. By following the steps outlined in this article, you can factor expressions with variables, fractions, exponents, negative numbers, decimals, mixed numbers, and imaginary numbers. Remember to combine like terms, simplify constant terms, and factor out the greatest common factor (GCF) to write the factored form. With practice and patience, you can master the art of factoring expressions and apply it to real-world problems.