Identify the pairs of terms

Oct 13, 2025 by leolavanderia.com.br 129 views

Introduction

Factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we will focus on factoring trinomials by grouping, which is a technique used to factorize trinomials of the form $ax^2 + bx + c$ . This method involves grouping the terms of the trinomial in a way that allows us to factor out common factors.

What is Factoring by Grouping?

Factoring by grouping is a technique used to factorize trinomials by grouping the terms in pairs. This method involves identifying two pairs of terms that can be factored out separately. The general form of a trinomial that can be factored by grouping is $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

How to Factor by Grouping

To factor a trinomial by grouping, we need to follow these steps:

Identify the pairs of terms: Identify two pairs of terms in the trinomial that can be factored out separately.
Factor out the common factors: Factor out the common factors from each pair of terms.
Combine the factors: Combine the factors from each pair to get the final factored form.

Example 1: Factoring by Grouping

Let's consider the trinomial $6w^2 - 40w + 66$ . We need to factor this trinomial by grouping.

Step 1: Identify the pairs of terms

The trinomial can be written as $(6w^2 - 40w) + (66)$ . We can identify two pairs of terms: $6w^2 - 40w$ and $66$ .

Step 2: Factor out the common factors

We can factor out $2w$ from the first pair of terms: $2w(3w - 20)$ . We can factor out $2$ from the second pair of terms: $2(33)$ .

Step 3: Combine the factors

We can combine the factors from each pair to get the final factored form: $2(3w - 11)(w - 3)$ .

Conclusion

In this article, we learned how to factor trinomials by grouping. We identified the pairs of terms, factored out the common factors, and combined the factors to get the final factored form. We also applied this technique to the trinomial $6w^2 - 40w + 66$ and obtained the factored form $2(3w - 11)(w - 3)$ .

Answer

The correct answer is B. $2(3w - 11)(w - 3)$ .

Discussion

Factoring by grouping is a powerful technique used to factorize trinomials. It involves identifying pairs of terms, factoring out common factors, and combining the factors to get the final factored form. This technique can be applied to a wide range of trinomials and is an essential tool in algebra.

Tips and Tricks

Identify the pairs of terms: The first step in factoring by grouping is to identify the pairs of terms. This can be done by looking for common factors or by using the distributive property.
Factor out the common factors: Once the pairs of terms are identified, the next step is to factor out the common factors. This can be done by looking for the greatest common factor (GCF) of the terms.
Combine the factors: The final step is to combine the factors from each pair to get the final factored form.

Common Mistakes

Not identifying the pairs of terms: One common mistake is not identifying the pairs of terms. This can lead to incorrect factoring.
Not factoring out the common factors: Another common mistake is not factoring out the common factors. This can also lead to incorrect factoring.
Not combining the factors: The final common mistake is not combining the factors. This can also lead to incorrect factoring.

Conclusion

In conclusion, factoring by grouping is a powerful technique used to factorize trinomials. It involves identifying pairs of terms, factoring out common factors, and combining the factors to get the final factored form. By following these steps and avoiding common mistakes, we can factor trinomials with ease.

Final Answer

Introduction

In our previous article, we learned how to factor trinomials by grouping. This technique involves identifying pairs of terms, factoring out common factors, and combining the factors to get the final factored form. In this article, we will answer some frequently asked questions about factoring by grouping.

Q&A

Q: What is factoring by grouping?

A: Factoring by grouping is a technique used to factorize trinomials by grouping the terms in pairs. This method involves identifying two pairs of terms that can be factored out separately.

Q: How do I identify the pairs of terms?

A: To identify the pairs of terms, look for common factors or use the distributive property. You can also try to factor out the greatest common factor (GCF) of the terms.

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

A: The greatest common factor (GCF) is the largest factor that divides all the terms of a trinomial. It is used to factor out common factors from each pair of terms.

Q: How do I factor out the common factors?

A: To factor out the common factors, look for the GCF of the terms and factor it out. You can also use the distributive property to factor out common factors.

Q: How do I combine the factors?

A: To combine the factors, multiply the factors from each pair to get the final factored form.

Q: What are some common mistakes to avoid?

A: Some common mistakes to avoid include not identifying the pairs of terms, not factoring out the common factors, and not combining the factors.

Q: Can I use factoring by grouping to factor all trinomials?

A: No, factoring by grouping can only be used to factor trinomials that can be written in the form $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I know if a trinomial can be factored by grouping?

A: To determine if a trinomial can be factored by grouping, look for two pairs of terms that can be factored out separately. If you can find two pairs of terms, then the trinomial can be factored by grouping.

Q: Can I use factoring by grouping to factor quadratic expressions?

A: Yes, factoring by grouping can be used to factor quadratic expressions. However, it is not the only method used to factor quadratic expressions.

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

A: Factoring by grouping has many real-world applications, including cryptography, coding theory, and computer science.

Conclusion

In conclusion, factoring by grouping is a powerful technique used to factorize trinomials. By understanding how to identify pairs of terms, factor out common factors, and combine the factors, you can factor trinomials with ease. Remember to avoid common mistakes and use factoring by grouping to factor trinomials that can be written in the form $ax^2 + bx + c$ .

Final Answer

The final answer is B. $2(3w - 11)(w - 3)$ .