Factor This Trinomial Completely:$2x^2 + 6x + 4$A. $2(x+2)(x+1$\] B. $2(x-2)(x-1$\] C. $2(x+2)(x-1$\] D. $2(x-2)(x+1$\]

Introduction

Factoring a trinomial 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 the trinomial $2x^2 + 6x + 4$. We will explore the different methods of factoring and provide a step-by-step guide on how to factor this trinomial completely.

What is a Trinomial?

A trinomial is a polynomial expression that consists of three terms. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In the given trinomial $2x^2 + 6x + 4$, $a = 2$, $b = 6$, and $c = 4$.

Methods of Factoring

There are several methods of factoring a trinomial, including:

• Factoring by Grouping: This method involves grouping the terms of the trinomial into two pairs and then factoring out the greatest common factor (GCF) from each pair.
• Factoring by Using the Quadratic Formula: This method involves using the quadratic formula to find the roots of the trinomial and then factoring the trinomial using the roots.
• Factoring by Using the FOIL Method: This method involves using the FOIL (First, Outer, Inner, Last) method to factor the trinomial.

Factoring the Trinomial $2x^2 + 6x + 4$

In this section, we will use the factoring by grouping method to factor the trinomial $2x^2 + 6x + 4$.

Step 1: Group the Terms

The first step in factoring the trinomial is to group the terms into two pairs. We can group the terms as follows:

$2x^2 + 6x + 4 = (2x^2 + 6x) + 4$

Step 2: Factor Out the GCF

The next step is to factor out the greatest common factor (GCF) from each pair. The GCF of $2x^2$ and $6x$ is $2x$, so we can factor out $2x$ from each pair:

$(2x^2 + 6x) + 4 = 2x(x + 3) + 4$

Step 3: Factor Out the GCF from the Remaining Term

The next step is to factor out the GCF from the remaining term. The GCF of $4$ is $4$, so we can factor out $4$ from the remaining term:

$2x(x + 3) + 4 = 2x(x + 3) + 4(1)$

Step 4: Factor the Trinomial

The final step is to factor the trinomial. We can factor the trinomial as follows:

$2x(x + 3) + 4(1) = 2(x + 2)(x + 1)$

Therefore, the factored form of the trinomial $2x^2 + 6x + 4$ is $2(x + 2)(x + 1)$.

Conclusion

In this article, we have explored the different methods of factoring a trinomial and provided a step-by-step guide on how to factor the trinomial $2x^2 + 6x + 4$. We have used the factoring by grouping method to factor the trinomial and have shown that the factored form of the trinomial is $2(x + 2)(x + 1)$. We hope that this article has provided a clear understanding of how to factor a trinomial and has helped to build a strong foundation in algebra.

Answer

The correct answer is:

A. $2(x+2)(x+1)$

Other Options

The other options are:

B. $2(x-2)(x-1)$

C. $2(x+2)(x-1)$

D. $2(x-2)(x+1)$

These options are incorrect because they do not factor the trinomial $2x^2 + 6x + 4$ correctly.

Final Thoughts

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Introduction

Factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In our previous article, we explored the different methods of factoring a trinomial and provided a step-by-step guide on how to factor the trinomial $2x^2 + 6x + 4$. In this article, we will answer some of the most frequently asked questions about factoring trinomials.

Q: What is the difference between factoring and simplifying a trinomial?

A: Factoring a trinomial involves expressing it as a product of two binomials, while simplifying a trinomial involves combining like terms to reduce the expression to its simplest form.

Q: How do I determine if a trinomial can be factored?

A: To determine if a trinomial can be factored, you need to check if the product of the coefficients of the two middle terms is equal to the constant term. If it is, then the trinomial can be factored.

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

A: The greatest common factor (GCF) of a trinomial is the largest expression that divides each term of the trinomial without leaving a remainder.

Q: How do I factor a trinomial using the factoring by grouping method?

A: To factor a trinomial using the factoring by grouping method, you need to group the terms into two pairs and then factor out the greatest common factor (GCF) from each pair.

Q: What is the difference between factoring by grouping and factoring by using the FOIL method?

A: Factoring by grouping involves grouping the terms into two pairs and then factoring out the greatest common factor (GCF) from each pair, while factoring by using the FOIL method involves using the FOIL (First, Outer, Inner, Last) method to factor the trinomial.

Q: How do I factor a trinomial using the FOIL method?

A: To factor a trinomial using the FOIL method, you need to multiply the first terms of the two binomials, then multiply the outer terms, then multiply the inner terms, and finally multiply the last terms. Then, you need to add or subtract the products to get the final factored form.

Q: What is the difference between factoring a trinomial and factoring a quadratic expression?

A: Factoring a trinomial involves expressing it as a product of two binomials, while factoring a quadratic expression involves expressing it as a product of two linear factors.

Q: How do I determine if a quadratic expression can be factored?

A: To determine if a quadratic expression can be factored, you need to check if the product of the coefficients of the two middle terms is equal to the constant term. If it is, then the quadratic expression can be factored.

Q: What is the greatest common factor (GCF) of a quadratic expression?

A: The greatest common factor (GCF) of a quadratic expression is the largest expression that divides each term of the quadratic expression without leaving a remainder.

Conclusion

In this article, we have answered some of the most frequently asked questions about factoring trinomials. We hope that this article has provided a clear understanding of how to factor a trinomial and has helped to build a strong foundation in algebra.

Final Thoughts

Factoring trinomials is an important concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we have explored the different methods of factoring a trinomial and have provided answers to some of the most frequently asked questions about factoring trinomials. We hope that this article has provided a clear understanding of how to factor a trinomial and has helped to build a strong foundation in algebra.