Factor The Trinomial Below: $14x^2 - 39x - 35$A. $(2x - 5)(7x + 7)$ B. \$(2x - 7)(7x + 5)$[/tex\] C. $(2x + 7)(7x - 5)$ D. $(2x + 5)(7x - 7)$

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 the trinomial $14x^2 - 39x - 35$ and explore the different methods and techniques used to factorize it. We will also discuss the importance of factoring trinomials in mathematics and provide examples of how it is applied in real-world scenarios.

What is Factoring a Trinomial?

Factoring a trinomial involves expressing it as a product of two binomials. A trinomial is a quadratic expression that consists of three terms, and factoring it involves finding two binomials whose product equals the original trinomial. Factoring trinomials is an essential skill in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions.

Methods of Factoring Trinomials

There are several methods of factoring trinomials, 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 Greatest Common Factor (GCF): This method involves factoring out the GCF from the trinomial, which is the largest factor that divides all three terms.
• Factoring by Using the Difference of Squares: This method involves factoring the trinomial as the difference of two squares, which is a special case of factoring.

Factoring the Trinomial $14x^2 - 39x - 35$

To factor the trinomial $14x^2 - 39x - 35$, we can use the method of factoring by grouping. This involves grouping the terms of the trinomial into two pairs and then factoring out the GCF from each pair.

Step 1: Group the Terms

The first step in factoring the trinomial is to group the terms into two pairs. We can do this by pairing the first and last terms, and the second and third terms.

$$14x^2 - 39x - 35 = (14x^2 - 35) - (39x)$$

Step 2: Factor Out the GCF

The next step is to factor out the GCF from each pair. The GCF of $14x^2$ and $-35$ is $7$, and the GCF of $-39x$ and $-35$ is $-7$.

$$14x^2 - 35 = 7(2x^2 - 5)$$

$$-39x = -7(3x)$$

Step 3: Combine the Factored Terms

The final step is to combine the factored terms to get the final factorization of the trinomial.

$$14x^2 - 39x - 35 = 7(2x^2 - 5) - 7(3x)$$

$$= 7(2x^2 - 5 - 3x)$$

$$= 7(2x^2 - 3x - 5)$$

However, this is not one of the answer choices. We need to try another method.

Step 2: Use the Factoring Formula

We can use the factoring formula to factor the trinomial. The factoring formula is:

$$ax^2 + bx + c = (mx + n)(px + q)$$

where $m$, $n$, $p$, and $q$ are constants.

We can use this formula to factor the trinomial $14x^2 - 39x - 35$.

$$14x^2 - 39x - 35 = (2x - 7)(7x + 5)$$

This is one of the answer choices.

Step 3: Check the Answer

To check the answer, we can multiply the two binomials together to get the original trinomial.

$$(2x - 7)(7x + 5) = 14x^2 - 39x - 35$$

This confirms that the correct factorization of the trinomial $14x^2 - 39x - 35$ is indeed $(2x - 7)(7x + 5)$.

Conclusion

Factoring trinomials is an essential skill in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we discussed the different methods of factoring trinomials, including factoring by grouping, factoring by using the greatest common factor (GCF), and factoring by using the difference of squares. We also provided a step-by-step guide on how to factor the trinomial $14x^2 - 39x - 35$ using the method of factoring by grouping and the factoring formula. The correct factorization of the trinomial $14x^2 - 39x - 35$ is indeed $(2x - 7)(7x + 5)$.

Answer

The correct answer is:

• B. $(2x - 7)(7x + 5)$

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 discussed the different methods of factoring trinomials, including factoring by grouping, factoring by using the greatest common factor (GCF), and factoring by using the difference of squares. In this article, we will provide a Q&A guide on factoring trinomials, covering common questions and topics related to this concept.

Q: What is factoring a trinomial?

A: Factoring a trinomial involves expressing it as a product of two binomials. A trinomial is a quadratic expression that consists of three terms, and factoring it involves finding two binomials whose product equals the original trinomial.

Q: Why is factoring trinomials important?

A: Factoring trinomials is an essential skill in algebra that has numerous applications in mathematics and real-world scenarios. By mastering the different methods of factoring trinomials, students can simplify complex expressions, solve equations, and analyze functions.

Q: What are the different methods of factoring trinomials?

A: There are several methods of factoring trinomials, 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 Greatest Common Factor (GCF): This method involves factoring out the GCF from the trinomial, which is the largest factor that divides all three terms.
• Factoring by Using the Difference of Squares: This method involves factoring the trinomial as the difference of two squares, which is a special case of factoring.

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

A: To factor a trinomial using the method of factoring by grouping, follow these steps:

1. Group the terms of the trinomial into two pairs.
2. Factor out the greatest common factor (GCF) from each pair.
3. Combine the factored terms to get the final factorization of the trinomial.

Q: How do I factor a trinomial using the factoring formula?

A: To factor a trinomial using the factoring formula, follow these steps:

1. Identify the coefficients of the trinomial.
2. Use the factoring formula to factor the trinomial.
3. Simplify the expression to get the final factorization of the trinomial.

Q: What is the factoring formula?

A: The factoring formula is:

$$ax^2 + bx + c = (mx + n)(px + q)$$

where $m$, $n$, $p$, and $q$ are constants.

Q: How do I check the answer when factoring a trinomial?

A: To check the answer when factoring a trinomial, multiply the two binomials together to get the original trinomial. If the result is equal to the original trinomial, then the factorization is correct.

Q: What are some common mistakes to avoid when factoring trinomials?

A: Some common mistakes to avoid when factoring trinomials include:

• Not factoring out the greatest common factor (GCF): Make sure to factor out the GCF from each pair of terms.
• Not using the correct method: Choose the correct method of factoring based on the type of trinomial.
• Not checking the answer: Always check the answer by multiplying the two binomials together.

Conclusion

Factoring trinomials is a fundamental concept in algebra that has numerous applications in mathematics and real-world scenarios. By mastering the different methods of factoring trinomials, students can simplify complex expressions, solve equations, and analyze functions. In this article, we provided a Q&A guide on factoring trinomials, covering common questions and topics related to this concept. We hope that this article has provided valuable insights and knowledge on factoring trinomials.

Final Thoughts

Factoring trinomials is a critical skill in algebra that requires practice and patience. By mastering the different methods of factoring trinomials, students can develop a deeper understanding of algebraic expressions and functions. We encourage students to practice factoring trinomials regularly to build their confidence and skills.