Factoring the Common Factor

Oct 13, 2025 by vinooalluri.com 51 views

Introduction

In algebra, factoring polynomials is an essential skill that helps us simplify complex expressions and solve equations. One method of factoring polynomials is by grouping, which involves rearranging the terms of a polynomial to facilitate factoring. In this article, we will explore how to determine the factors of a given polynomial by grouping.

What is Grouping in Factoring?

Grouping is a factoring technique that involves rearranging the terms of a polynomial to create two or more groups of terms that can be factored separately. This method is particularly useful for polynomials with four or more terms. By grouping, we can often factor out common factors from each group, making it easier to find the factors of the polynomial.

The Given Polynomial

The given polynomial is $$x^3 + 4x^2 + 5x + 20$$. Our goal is to determine the factors of this polynomial by grouping.

Option A: $x(x^2 + 4) + 5(x^2 + 4)$

Let's examine Option A: $x(x^2 + 4) + 5(x^2 + 4)$ . To determine if this is the correct grouping, we need to check if the terms in each group have a common factor.

# Option A: $x(x^2 + 4) + 5(x^2 + 4)$
Factoring the Groups
 $x(x^2 + 4) + 5(x^2 + 4)$ 
Factoring the Common Factor
 $(x + 5)(x^2 + 4)$

As we can see, the terms in each group have a common factor of $(x^2 + 4)$ . Factoring out this common factor, we get $(x + 5)(x^2 + 4)$ .

Option B: $x^2(x + 4) + 5(x + 4)$

Let's examine Option B: $x^2(x + 4) + 5(x + 4)$ . To determine if this is the correct grouping, we need to check if the terms in each group have a common factor.

# Option B: $x^2(x + 4) + 5(x + 4)$
Factoring the Groups
 $x^2(x + 4) + 5(x + 4)$ 
Factoring the Common Factor
 $(x^2 + 5)(x + 4)$

As we can see, the terms in each group have a common factor of $(x + 4)$ . Factoring out this common factor, we get $(x^2 + 5)(x + 4)$ .

Option C: $x^2(x + 5) + 4(x + 5)$

Let's examine Option C: $x^2(x + 5) + 4(x + 5)$ . To determine if this is the correct grouping, we need to check if the terms in each group have a common factor.

# Option C: $x^2(x + 5) + 4(x + 5)$
Factoring the Groups
 $x^2(x + 5) + 4(x + 5)$ 
Factoring the Common Factor
 $(x^2 + 4)(x + 5)$

As we can see, the terms in each group have a common factor of $(x + 5)$ . Factoring out this common factor, we get $(x^2 + 4)(x + 5)$ .

Option D: $x(x^2 + 5) + 4x(x^2 + 5)$

Let's examine Option D: $x(x^2 + 5) + 4x(x^2 + 5)$ . To determine if this is the correct grouping, we need to check if the terms in each group have a common factor.

# Option D: $x(x^2 + 5) + 4x(x^2 + 5)$
Factoring the Groups
 $x(x^2 + 5) + 4x(x^2 + 5)$ 
Factoring the Common Factor
 $x(x^2 + 5)(x + 4)$

As we can see, the terms in each group have a common factor of $(x^2 + 5)$ . Factoring out this common factor, we get $x(x^2 + 5)(x + 4)$ .

Conclusion

In conclusion, the correct grouping for the given polynomial is Option A: $x(x^2 + 4) + 5(x^2 + 4)$ . By factoring out the common factor $(x^2 + 4)$ , we get $(x + 5)(x^2 + 4)$ , which is the factorization of the given polynomial.

Final Answer

Introduction

In our previous article, we explored how to determine the factors of a polynomial by grouping. Grouping is a factoring technique that involves rearranging the terms of a polynomial to create two or more groups of terms that can be factored separately. In this article, we will answer some frequently asked questions about determining factors of a polynomial by grouping.

Q: What is the purpose of grouping in factoring?

A: The purpose of grouping in factoring is to rearrange the terms of a polynomial to create two or more groups of terms that can be factored separately. This method is particularly useful for polynomials with four or more terms.

Q: How do I determine the correct grouping for a polynomial?

A: To determine the correct grouping for a polynomial, you need to examine the terms of the polynomial and look for common factors. You can try different groupings and see if the terms in each group have a common factor.

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

A: Some common mistakes to avoid when grouping include:

Not factoring out the common factor correctly
Not checking if the terms in each group have a common factor
Not considering all possible groupings

Q: Can I use grouping to factor polynomials with more than four terms?

A: Yes, you can use grouping to factor polynomials with more than four terms. However, you may need to use multiple groupings to factor the polynomial completely.

Q: How do I know if I have factored the polynomial correctly?

A: To know if you have factored the polynomial correctly, you need to check if the factors you have obtained are correct. You can do this by multiplying the factors together and checking if you get the original polynomial.

Q: What are some examples of polynomials that can be factored using grouping?

A: Some examples of polynomials that can be factored using grouping include:

$x^3 + 4x^2 + 5x + 20$
$x^4 + 2x^3 + 3x^2 + 4x + 5$
$x^5 + 3x^4 + 2x^3 + x^2 + 4x + 5$

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

A: Yes, you can use grouping to factor polynomials with negative coefficients. However, you may need to use the distributive property to factor out the negative coefficients.

Q: How do I apply the distributive property when factoring polynomials with negative coefficients?

A: To apply the distributive property when factoring polynomials with negative coefficients, you need to multiply each term in the polynomial by the negative coefficient. For example, if you have the polynomial $-x^2 + 3x - 4$ , you can factor out the negative coefficient by multiplying each term by $-1$ to get $-x^2 + 3x - 4 = -(x^2 - 3x + 4)$ .

Conclusion

In conclusion, determining factors of a polynomial by grouping is a useful technique that can help you simplify complex expressions and solve equations. By following the steps outlined in this article, you can learn how to determine the factors of a polynomial by grouping and apply this technique to a variety of problems.

Final Tips

Always check if the terms in each group have a common factor before factoring out the common factor.
Consider all possible groupings when factoring a polynomial.
Use the distributive property to factor out negative coefficients.
Check if the factors you have obtained are correct by multiplying them together and checking if you get the original polynomial.