Factor out the common binomial factor

Oct 13, 2025 by magicalceylontours.com 36 views

Introduction

In algebra, factoring is a crucial technique used to simplify and solve polynomial expressions. One of the most effective methods of factoring is the factor by grouping technique. This method involves grouping the terms of a polynomial expression in a way that allows us to factor out common factors. In this article, we will delve into the concept of factor by grouping and provide a step-by-step guide on how to apply it to factor the given polynomial expression: $x^3 - 6x^2 + 2x - 12$ .

What is Factor by Grouping?

Factor by grouping is a factoring technique used to factor quadratic and higher-degree polynomial expressions. It involves grouping the terms of the polynomial expression in pairs and then factoring out common factors from each pair. This method is particularly useful when the polynomial expression has no common factors that can be factored out directly.

How to Factor by Grouping

To factor a polynomial expression using the factor by grouping technique, follow these steps:

Group the terms: Group the terms of the polynomial expression in pairs. For example, in the expression $x^3 - 6x^2 + 2x - 12$ , we can group the first two terms and the last two terms as follows: $(x^3 - 6x^2) + (2x - 12)$ .
Factor out common factors: Factor out common factors from each pair of terms. In the expression $(x^3 - 6x^2) + (2x - 12)$ , we can factor out $x^2$ from the first pair of terms and $2$ from the second pair of terms as follows: $x^2(x - 6) + 2(x - 6)$ .
Factor out the common binomial factor: Factor out the common binomial factor from the two pairs of terms. In the expression $x^2(x - 6) + 2(x - 6)$ , we can factor out the common binomial factor $(x - 6)$ as follows: $(x^2 + 2)(x - 6)$ .

Step-by-Step Solution

Now, let's apply the factor by grouping technique to factor the given polynomial expression: $x^3 - 6x^2 + 2x - 12$ .

Step 1: Group the terms

Group the terms of the polynomial expression in pairs as follows: $(x^3 - 6x^2) + (2x - 12)$ .

Step 2: Factor out common factors

Factor out common factors from each pair of terms as follows: $x^2(x - 6) + 2(x - 6)$ .

Step 3: Factor out the common binomial factor

Factor out the common binomial factor from the two pairs of terms as follows: $(x^2 + 2)(x - 6)$ .

Conclusion

In this article, we have discussed the concept of factor by grouping and provided a step-by-step guide on how to apply it to factor the given polynomial expression: $x^3 - 6x^2 + 2x - 12$ . By following the steps outlined in this article, you should be able to factor polynomial expressions using the factor by grouping technique. Remember to group the terms, factor out common factors, and factor out the common binomial factor to simplify and solve polynomial expressions.

Example Problems

Problem 1

Factor the polynomial expression: $x^2 + 5x + 6$ .

Solution

To factor the polynomial expression $x^2 + 5x + 6$ , we can group the terms as follows: $(x^2 + 4x) + (x + 6)$ . Then, we can factor out common factors from each pair of terms as follows: $x(x + 4) + 1(x + 6)$ . Finally, we can factor out the common binomial factor as follows: $(x + 1)(x + 6)$ .

Problem 2

Factor the polynomial expression: $x^3 + 2x^2 - 7x - 14$ .

Solution

To factor the polynomial expression $x^3 + 2x^2 - 7x - 14$ , we can group the terms as follows: $(x^3 + 2x^2) + (-7x - 14)$ . Then, we can factor out common factors from each pair of terms as follows: $x^2(x + 2) - 7(x + 2)$ . Finally, we can factor out the common binomial factor as follows: $(x^2 - 7)(x + 2)$ .

Tips and Tricks

Group the terms carefully: When grouping the terms, make sure to group them in pairs that have common factors.
Factor out common factors: When factoring out common factors, make sure to factor out the greatest common factor (GCF) from each pair of terms.
Factor out the common binomial factor: When factoring out the common binomial factor, make sure to factor out the binomial factor that is common to both pairs of terms.

Conclusion

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about factor by grouping.

Q: What is factor by grouping?

A: Factor by grouping is a factoring technique used to simplify and solve polynomial expressions. It involves grouping the terms of a polynomial expression in pairs and then factoring out common factors from each pair.

Q: When should I use factor by grouping?

A: You should use factor by grouping when the polynomial expression has no common factors that can be factored out directly. This method is particularly useful when the polynomial expression has multiple terms with common factors.

Q: How do I group the terms in a polynomial expression?

A: To group the terms in a polynomial expression, you should pair the terms in a way that allows you to factor out common factors. For example, in the expression $x^3 - 6x^2 + 2x - 12$ , you can group the first two terms and the last two terms as follows: $(x^3 - 6x^2) + (2x - 12)$ .

Q: How do I factor out common factors from each pair of terms?

A: To factor out common factors from each pair of terms, you should identify the greatest common factor (GCF) of each pair of terms and factor it out. For example, in the expression $x^3 - 6x^2 + 2x - 12$ , you can factor out $x^2$ from the first pair of terms and $2$ from the second pair of terms as follows: $x^2(x - 6) + 2(x - 6)$ .

Q: How do I factor out the common binomial factor?

A: To factor out the common binomial factor, you should identify the binomial factor that is common to both pairs of terms and factor it out. For example, in the expression $x^2(x - 6) + 2(x - 6)$ , you can factor out the common binomial factor $(x - 6)$ as follows: $(x^2 + 2)(x - 6)$ .

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

A: Some common mistakes to avoid when using factor by grouping include:

Not grouping the terms correctly: Make sure to group the terms in a way that allows you to factor out common factors.
Not factoring out the greatest common factor (GCF): Make sure to factor out the GCF from each pair of terms.
Not factoring out the common binomial factor: Make sure to factor out the binomial factor that is common to both pairs of terms.

Q: How do I know if a polynomial expression can be factored using factor by grouping?

A: You can determine if a polynomial expression can be factored using factor by grouping by looking for common factors among the terms. If the terms have common factors, you can try grouping the terms and factoring out the common factors.

Q: What are some examples of polynomial expressions that can be factored using factor by grouping?

A: Some examples of polynomial expressions that can be factored using factor by grouping include:

$x^3 - 6x^2 + 2x - 12$
$x^2 + 5x + 6$
$x^3 + 2x^2 - 7x - 14$

Conclusion

In conclusion, factor by grouping is a powerful technique used to simplify and solve polynomial expressions. By following the steps outlined in this article, you should be able to factor polynomial expressions using the factor by grouping technique. Remember to group the terms, factor out common factors, and factor out the common binomial factor to simplify and solve polynomial expressions.

Additional Resources

Factor by grouping worksheet: Try practicing factor by grouping with this worksheet.
Factor by grouping video tutorial: Watch this video tutorial for a step-by-step guide on how to factor by grouping.
Factor by grouping practice problems: Try solving these practice problems to test your understanding of factor by grouping.

Common Misconceptions

Factor by grouping only works for quadratic expressions: Factor by grouping can be used to factor polynomial expressions of any degree, not just quadratic expressions.
Factor by grouping is only used for simple polynomial expressions: Factor by grouping can be used to factor complex polynomial expressions, not just simple ones.
Factor by grouping is a difficult technique to learn: Factor by grouping is a relatively simple technique to learn, and with practice, you should be able to master it.