Factor The Polynomial By Grouping.${ 2a^2 - 12b + 8ab - 3a = }$

Introduction

In algebra, factoring polynomials is a crucial skill that helps us simplify complex expressions and solve equations. One of the techniques used to factor polynomials is the grouping method. This method involves grouping terms in a polynomial in a way that allows us to factor out common factors. In this article, we will explore how to factor a polynomial by grouping, using the given polynomial as an example.

The Given Polynomial

The given polynomial is:

${ 2a^2 - 12b + 8ab - 3a }$

Our goal is to factor this polynomial by grouping.

Step 1: Identify the Terms

To factor the polynomial by grouping, we need to identify the terms that can be grouped together. Let's examine the given polynomial:

${ 2a^2 - 12b + 8ab - 3a }$

We can see that the polynomial has four terms: $2a^2$, $-12b$, $8ab$, and $-3a$.

Step 2: Group the Terms

Now, let's group the terms in a way that allows us to factor out common factors. We can group the first two terms together and the last two terms together:

${ (2a^2 - 12b) + (8ab - 3a) }$

Step 3: Factor Out Common Factors

Now, let's factor out common factors from each group:

${ 2a(a - 6b) + a(8b - 3) }$

We can see that both groups have a common factor of $a$. Let's factor out $a$ from each group:

${ a(2a - 12b) + a(8b - 3) }$

Step 4: Factor Out the Common Binomial Factor

Now, let's factor out the common binomial factor $(2a - 12b)$ from the first group and $(8b - 3)$ from the second group:

${ a((2a - 12b) + (8b - 3)) }$

We can simplify the expression inside the parentheses:

${ a(2a - 12b + 8b - 3) }$

${ a(2a - 4b - 3) }$

The Final Answer

Therefore, the factored form of the polynomial is:

${ a(2a - 4b - 3) }$

Conclusion

In this article, we learned how to factor a polynomial by grouping. We identified the terms, grouped them together, factored out common factors, and finally factored out the common binomial factor. This technique is useful for simplifying complex expressions and solving equations.

Example Problems

Here are some example problems that you can try to practice factoring polynomials by grouping:

1. Factor the polynomial: $3x^2 + 6x + 9x + 18$
2. Factor the polynomial: $2y^2 - 4y + 6y - 12$
3. Factor the polynomial: $4z^2 + 8z + 12z + 24$

Tips and Tricks

Here are some tips and tricks to help you factor polynomials by grouping:

1. Identify the terms that can be grouped together.
2. Factor out common factors from each group.
3. Factor out the common binomial factor from each group.
4. Simplify the expression inside the parentheses.

Introduction

In our previous article, we learned how to factor a polynomial by grouping. This technique is useful for simplifying complex expressions and solving equations. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will provide a Q&A section to help you understand the concept of factoring polynomials by grouping.

Q: What is the first step in factoring a polynomial by grouping?

A: The first step in factoring a polynomial by grouping is to identify the terms that can be grouped together. This involves examining the polynomial and grouping the terms in a way that allows us to factor out common factors.

Q: How do I identify the terms that can be grouped together?

A: To identify the terms that can be grouped together, you need to examine the polynomial and look for common factors. You can group the terms in a way that allows you to factor out common factors, such as grouping the first two terms together and the last two terms together.

Q: What is the next step after identifying the terms that can be grouped together?

A: After identifying the terms that can be grouped together, the next step is to factor out common factors from each group. This involves factoring out common factors from each group, such as factoring out a common binomial factor.

Q: How do I factor out common factors from each group?

A: To factor out common factors from each group, you need to examine each group and look for common factors. You can factor out common factors from each group by dividing each term in the group by the common factor.

Q: What is the final step in factoring a polynomial by grouping?

A: The final step in factoring a polynomial by grouping is to simplify the expression inside the parentheses. This involves combining like terms and simplifying the expression.

Q: Can I factor a polynomial by grouping if it has more than four terms?

A: Yes, you can factor a polynomial by grouping if it has more than four terms. However, you need to group the terms in a way that allows you to factor out common factors.

Q: How do I know if I have factored a polynomial correctly by grouping?

A: To know if you have factored a polynomial correctly by grouping, you need to check your work by multiplying the factors together and simplifying the expression. If the result is the original polynomial, then you have factored it correctly.

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

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

• Not identifying the terms that can be grouped together
• Not factoring out common factors from each group
• Not simplifying the expression inside the parentheses
• Not checking your work by multiplying the factors together and simplifying the expression

Q: Can I use the grouping method to factor polynomials with negative coefficients?

A: Yes, you can use the grouping method to factor polynomials with negative coefficients. However, you need to be careful when factoring out common factors, as the negative sign may affect the result.

Q: Can I use the grouping method to factor polynomials with fractional coefficients?

A: Yes, you can use the grouping method to factor polynomials with fractional coefficients. However, you need to be careful when factoring out common factors, as the fractional coefficient may affect the result.

Conclusion

In this article, we provided a Q&A section to help you understand the concept of factoring polynomials by grouping. We answered common questions and provided tips and tricks to help you master the technique. By following these steps and tips, you can factor polynomials by grouping with ease and simplify complex expressions with confidence.

Example Problems

Here are some example problems that you can try to practice factoring polynomials by grouping:

1. Factor the polynomial: $3x^2 + 6x + 9x + 18$
2. Factor the polynomial: $2y^2 - 4y + 6y - 12$
3. Factor the polynomial: $4z^2 + 8z + 12z + 24$

Tips and Tricks

Here are some tips and tricks to help you factor polynomials by grouping:

1. Identify the terms that can be grouped together.
2. Factor out common factors from each group.
3. Factor out the common binomial factor from each group.
4. Simplify the expression inside the parentheses.
5. Check your work by multiplying the factors together and simplifying the expression.

By following these steps and tips, you can master the technique of factoring polynomials by grouping and simplify complex expressions with ease.