Factor The Following Trinomial:${ 5x^2 + 2x - 7 }$ { (5x + \text{[?]})(x - \text{[?]}) \}

Mar 13, 2025 by ADMIN 92 views

Factoring the Trinomial: A Step-by-Step Guide

=====================================================

Introduction

Factoring a trinomial is an essential skill in algebra, and it can be a bit challenging, especially when dealing with complex expressions. In this article, we will focus on factoring the trinomial $5x^2 + 2x - 7$ into the product of two binomials. We will use the method of factoring by grouping, which is a powerful technique for factoring trinomials.

Understanding the Trinomial

Before we dive into the factoring process, let's take a closer look at the trinomial $5x^2 + 2x - 7$ . This trinomial has three terms: $5x^2$ , $2x$ , and $-7$ . The first term, $5x^2$ , is a quadratic term, while the second term, $2x$ , is a linear term. The third term, $-7$ , is a constant term.

Factoring by Grouping

Factoring by grouping is a technique used to factor trinomials into the product of two binomials. This method involves grouping the terms of the trinomial into two pairs and then factoring out the greatest common factor (GCF) from each pair.

Step 1: Group the Terms

To factor the trinomial $5x^2 + 2x - 7$ by grouping, we need to group the terms into two pairs. We can do this by pairing the first two terms, $5x^2$ and $2x$ , and the last two terms, $2x$ and $-7$ . However, we notice that the first two terms do not have a common factor, so we will pair the first term with the last term and the second term with the second term.

Step 2: Factor Out the GCF

Now that we have grouped the terms, we need to factor out the greatest common factor (GCF) from each pair. The GCF of $5x^2$ and $-7$ is $1$ , and the GCF of $2x$ and $-7$ is $-1$ . However, we can factor out a common factor of $-1$ from the second pair.

Step 3: Rewrite the Trinomial

After factoring out the GCF from each pair, we can rewrite the trinomial as follows:

5x^2 + 2x - 7 = (5x^2 - 7) + (2x)

Step 4: Factor the First Pair

Now that we have rewritten the trinomial, we can factor the first pair, $5x^2 - 7$ . We can factor out a common factor of $5x^2$ from this pair.

Step 5: Factor the Second Pair

Next, we can factor the second pair, $2x$ . We can factor out a common factor of $2x$ from this pair.

Step 6: Combine the Factors

Finally, we can combine the factors from each pair to get the final factored form of the trinomial.

The Final Answer

After following the steps outlined above, we can factor the trinomial $5x^2 + 2x - 7$ into the product of two binomials as follows:

(5x + 1)(x - 7)

Conclusion

Factoring a trinomial can be a challenging task, but with the right technique and practice, it can become second nature. In this article, we used the method of factoring by grouping to factor the trinomial $5x^2 + 2x - 7$ into the product of two binomials. We hope that this article has provided you with a clear understanding of how to factor a trinomial using this method.

Frequently Asked Questions

Q: What is factoring by grouping?

A: Factoring by grouping is a technique used to factor trinomials into the product of two binomials. This method involves grouping the terms of the trinomial into two pairs and then factoring out the greatest common factor (GCF) from each pair.

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, 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 greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides each term of a polynomial.

Q: How do I find the greatest common factor (GCF) of two terms?

A: To find the greatest common factor (GCF) of two terms, you need to list the factors of each term and then find the largest factor that is common to both terms.

Tips and Tricks

Tip 1: Use the method of factoring by grouping to factor trinomials.

The method of factoring by grouping is a powerful technique for factoring trinomials. It involves grouping the terms of the trinomial into two pairs and then factoring out the greatest common factor (GCF) from each pair.

Tip 2: Factor out the greatest common factor (GCF) from each pair.

When factoring by grouping, it is essential to factor out the greatest common factor (GCF) from each pair. This will help you to simplify the trinomial and make it easier to factor.

Tip 3: Use the distributive property to expand the factored form.

When you have factored a trinomial using the method of factoring by grouping, you can use the distributive property to expand the factored form and verify that it is equivalent to the original trinomial.

Resources

Online Resources

Khan Academy: Factoring Trinomials
Mathway: Factoring Trinomials
Wolfram Alpha: Factoring Trinomials

Books

"Algebra" by Michael Artin
"College Algebra" by James Stewart
"Algebra and Trigonometry" by Michael Sullivan

Videos

Factoring Trinomials by Khan Academy
Factoring Trinomials by Mathway
Factoring Trinomials by Wolfram Alpha

=====================================

Introduction

Factoring trinomials can be a challenging task, but with the right techniques and practice, it can become second nature. In this article, we will provide a comprehensive Q&A guide to help you understand the concept of factoring trinomials and how to apply it to different types of trinomials.

Q&A

Q: What is a trinomial?

A: A trinomial is a polynomial expression that consists of three terms.

Q: What is factoring a trinomial?

A: Factoring a trinomial is the process of expressing it as the product of two binomials.

Q: What are the different methods of factoring trinomials?

A: There are several methods of factoring trinomials, including:

Factoring by grouping
Factoring by using the quadratic formula
Factoring by using the difference of squares formula
Factoring by using the sum and difference of cubes formula

Q: What is the method of factoring by grouping?

A: The method of factoring by grouping involves grouping the terms of the trinomial into two pairs and then factoring out the greatest common factor (GCF) from each pair.

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, you need to:

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

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

A: The greatest common factor (GCF) is the largest factor that divides each term of a polynomial.

Q: How do I find the greatest common factor (GCF) of two terms?

A: To find the greatest common factor (GCF) of two terms, you need to list the factors of each term and then find the largest factor that is common to both terms.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations of the form ax^2 + bx + c = 0.

Q: How do I use the quadratic formula to factor a trinomial?

A: To use the quadratic formula to factor a trinomial, you need to:

Write the trinomial in the form ax^2 + bx + c = 0.
Plug the values of a, b, and c into the quadratic formula.
Simplify the expression to get the final factored form.

Q: What is the difference of squares formula?

A: The difference of squares formula is a formula used to factor expressions of the form a^2 - b^2.

Q: How do I use the difference of squares formula to factor a trinomial?

A: To use the difference of squares formula to factor a trinomial, you need to:

Write the trinomial in the form a^2 - b^2.
Plug the values of a and b into the difference of squares formula.
Simplify the expression to get the final factored form.

Q: What is the sum and difference of cubes formula?

A: The sum and difference of cubes formula is a formula used to factor expressions of the form a^3 + b^3 and a^3 - b^3.

Q: How do I use the sum and difference of cubes formula to factor a trinomial?

A: To use the sum and difference of cubes formula to factor a trinomial, you need to:

Write the trinomial in the form a^3 + b^3 or a^3 - b^3.
Plug the values of a and b into the sum and difference of cubes formula.
Simplify the expression to get the final factored form.