Use The Drop-down Menus To Complete The Statements About Factoring $14x^2 + 6x - 7x - 3$ By Grouping.1. The GCF Of The Group $(14x^2 - 7x$\] Is $\square$.2. The GCF Of The Group $(6x - 3$\] Is $\square$.3.

Introduction

Factoring by grouping is a powerful technique used to simplify algebraic expressions. It involves grouping terms in pairs and factoring out the greatest common factor (GCF) from each group. In this article, we will explore how to factor the expression $14x^2 + 6x - 7x - 3$ by grouping.

Step 1: Identify the GCF of the First Group

To factor the expression by grouping, we need to identify the GCF of the first group, which is $(14x^2 - 7x)$. The GCF is the largest expression that divides both terms in the group without leaving a remainder.

• The GCF of the group $(14x^2 - 7x)$ is $\boxed{7x}$.

Step 2: Identify the GCF of the Second Group

Next, we need to identify the GCF of the second group, which is $(6x - 3)$. Again, the GCF is the largest expression that divides both terms in the group without leaving a remainder.

• The GCF of the group $(6x - 3)$ is $\boxed{3}$.

Step 3: Factor Out the GCF from Each Group

Now that we have identified the GCF of each group, we can factor out the GCF from each group.

• Factoring out the GCF from the first group: $14x^2 - 7x = 7x(2x - 1)$.
• Factoring out the GCF from the second group: $6x - 3 = 3(2x - 1)$.

Step 4: Rewrite the Expression

Now that we have factored out the GCF from each group, we can rewrite the original expression as a product of two binomials.

• $14x^2 + 6x - 7x - 3 = 7x(2x - 1) + 3(2x - 1)$.

Step 5: Factor Out the Common Binomial

Finally, we can factor out the common binomial $(2x - 1)$ from both terms.

• $7x(2x - 1) + 3(2x - 1) = (2x - 1)(7x + 3)$.

Conclusion

In this article, we have shown how to factor the expression $14x^2 + 6x - 7x - 3$ by grouping. We identified the GCF of each group, factored out the GCF from each group, and finally factored out the common binomial. This technique is a powerful tool for simplifying algebraic expressions and can be used to solve a wide range of problems.

Example Problems

Here are a few example problems that demonstrate how to factor by grouping:

Example 1

Factor the expression $12x^2 + 9x - 4x - 3$ by grouping.

• Step 1: Identify the GCF of the first group: $(12x^2 - 4x) = 4x(3x - 1)$.
• Step 2: Identify the GCF of the second group: $(9x - 3) = 3(3x - 1)$.
• Step 3: Factor out the GCF from each group: $4x(3x - 1) + 3(3x - 1) = (3x - 1)(4x + 3)$.
• Step 4: Rewrite the expression: $12x^2 + 9x - 4x - 3 = (3x - 1)(4x + 3)$.

Example 2

Factor the expression $15x^2 + 10x - 5x - 3$ by grouping.

• Step 1: Identify the GCF of the first group: $(15x^2 - 5x) = 5x(3x - 1)$.
• Step 2: Identify the GCF of the second group: $(10x - 3) = 1(10x - 3)$.
• Step 3: Factor out the GCF from each group: $5x(3x - 1) + 1(10x - 3) = (3x - 1)(5x + 1)$.
• Step 4: Rewrite the expression: $15x^2 + 10x - 5x - 3 = (3x - 1)(5x + 1)$.

Example 3

Factor the expression $20x^2 + 12x - 8x - 6$ by grouping.

• Step 1: Identify the GCF of the first group: $(20x^2 - 8x) = 4x(5x - 2)$.
• Step 2: Identify the GCF of the second group: $(12x - 6) = 6(2x - 1)$.
• Step 3: Factor out the GCF from each group: $4x(5x - 2) + 6(2x - 1) = (5x - 2)(4x + 6)$.
• Step 4: Rewrite the expression: $20x^2 + 12x - 8x - 6 = (5x - 2)(4x + 6)$.

Tips and Tricks

Here are a few tips and tricks to help you master the art of factoring by grouping:

• Pay attention to the signs: When factoring by grouping, pay close attention to the signs of the terms in each group. This will help you identify the GCF and factor out the common binomial.
• Use the distributive property: The distributive property states that $a(b + c) = ab + ac$. Use this property to expand the expression and identify the GCF.
• Look for common factors: When factoring by grouping, look for common factors in each group. This will help you identify the GCF and factor out the common binomial.
• Practice, practice, practice: Factoring by grouping is a skill that takes practice to develop. Make sure to practice regularly to become proficient in this technique.

Conclusion

Q: What is factoring by grouping?

A: Factoring by grouping is a technique used to simplify algebraic expressions by grouping terms in pairs and factoring out the greatest common factor (GCF) from each group.

Q: How do I identify the GCF of a group?

A: To identify the GCF of a group, look for the largest expression that divides both terms in the group without leaving a remainder. You can use the distributive property to expand the expression and identify the GCF.

Q: What is the distributive property?

A: The distributive property states that $a(b + c) = ab + ac$. This property can be used to expand an expression and identify the GCF.

Q: How do I factor out the GCF from each group?

A: To factor out the GCF from each group, divide each term in the group by the GCF. This will give you a factored form of the expression.

Q: What is the common binomial?

A: The common binomial is the expression that is factored out from both groups. It is the expression that is common to both groups.

Q: How do I rewrite the expression as a product of two binomials?

A: To rewrite the expression as a product of two binomials, multiply the factored form of each group together. This will give you the final factored form of the expression.

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

A: Some common mistakes to avoid when factoring by grouping include:

• Not identifying the GCF correctly: Make sure to identify the GCF correctly by looking for the largest expression that divides both terms in the group without leaving a remainder.
• Not factoring out the GCF correctly: Make sure to factor out the GCF correctly by dividing each term in the group by the GCF.
• Not rewriting the expression correctly: Make sure to rewrite the expression correctly by multiplying the factored form of each group together.

Q: How do I practice factoring by grouping?

A: To practice factoring by grouping, try the following:

• Start with simple expressions: Begin with simple expressions and gradually move on to more complex ones.
• Use online resources: There are many online resources available that can help you practice factoring by grouping.
• Work with a partner: Working with a partner can help you stay motivated and get feedback on your work.

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

A: Factoring by grouping has many real-world applications, including:

• Simplifying algebraic expressions: Factoring by grouping can be used to simplify algebraic expressions, making them easier to work with.
• Solving equations: Factoring by grouping can be used to solve equations, making it a useful tool for solving problems in mathematics and science.
• Modeling real-world situations: Factoring by grouping can be used to model real-world situations, making it a useful tool for solving problems in fields such as economics and engineering.

Q: How do I know if I am doing factoring by grouping correctly?

A: To know if you are doing factoring by grouping correctly, make sure to:

• Check your work: Check your work carefully to make sure that you have factored out the GCF correctly and rewritten the expression correctly.
• Use online resources: Use online resources to check your work and get feedback on your factoring by grouping skills.
• Practice regularly: Practice factoring by grouping regularly to develop your skills and build your confidence.

Conclusion

In conclusion, factoring by grouping is a powerful technique used to simplify algebraic expressions. By identifying the GCF of each group, factoring out the GCF from each group, and finally factoring out the common binomial, we can rewrite the original expression as a product of two binomials. With practice and patience, you can master the art of factoring by grouping and become proficient in this technique.