Factor The Expression:$\[ 7y^2 - 13y - 2 \\]

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. This technique is essential in solving quadratic equations, simplifying expressions, and understanding the properties of quadratic functions. In this article, we will focus on factoring the expression $7y^2 - 13y - 2$ using various methods.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The expression $7y^2 - 13y - 2$ is a quadratic expression in the variable $y$.

Factoring by Grouping

One method of factoring quadratic expressions is by grouping. This method involves grouping the terms of the expression into two pairs and then factoring out the greatest common factor (GCF) from each pair.

Step 1: Group the terms

The expression $7y^2 - 13y - 2$ can be grouped as follows:

$7y^2 - 13y - 2 = (7y^2 - 13y) - 2$

Step 2: Factor out the GCF from each pair

The GCF of $7y^2$ and $-13y$ is $7y$. The GCF of $-2$ is $-2$.

$(7y^2 - 13y) - 2 = 7y(y - \frac{13}{7}) - 2$

Step 3: Factor out the common binomial factor

The common binomial factor is $7y - 2$.

$7y(y - \frac{13}{7}) - 2 = (7y - 2)(y - \frac{13}{7})$

Step 4: Simplify the expression

The expression can be simplified by multiplying the two binomials.

$(7y - 2)(y - \frac{13}{7}) = 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

$= 7y^2 - \frac{91}{7}y - 2y + \frac{26}{7}$

Introduction

Factoring quadratic expressions 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 method of factoring by grouping. In this article, we will provide a Q&A guide to help you understand the concept of factoring quadratic expressions.

Q: What is a quadratic expression?

A: A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What is factoring?

A: Factoring is the process of expressing a quadratic expression as a product of two binomials. This involves finding two binomials whose product is equal to the original quadratic expression.

Q: What are the different methods of factoring quadratic expressions?

A: There are several methods of factoring quadratic expressions, including:

• Factoring by grouping
• Factoring by using the greatest common factor (GCF)
• Factoring by using the difference of squares
• Factoring by using the sum and difference of cubes

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

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

1. Group the terms of the expression into two pairs.
2. Factor out the greatest common factor (GCF) from each pair.
3. Factor out the common binomial factor from the two pairs.

Q: How do I factor a quadratic expression using the method of factoring by using the greatest common factor (GCF)?

A: To factor a quadratic expression using the method of factoring by using the greatest common factor (GCF), follow these steps:

1. Identify the greatest common factor (GCF) of the terms of the expression.
2. Factor out the GCF from each term.
3. Write the expression as a product of the GCF and the remaining terms.

Q: How do I factor a quadratic expression using the method of factoring by using the difference of squares?

A: To factor a quadratic expression using the method of factoring by using the difference of squares, follow these steps:

1. Identify the difference of squares in the expression.
2. Factor the difference of squares using the formula $(a-b)(a+b)$.
3. Write the expression as a product of the two binomials.

Q: How do I factor a quadratic expression using the method of factoring by using the sum and difference of cubes?

A: To factor a quadratic expression using the method of factoring by using the sum and difference of cubes, follow these steps:

1. Identify the sum or difference of cubes in the expression.
2. Factor the sum or difference of cubes using the formula $(a+b)(a^2-ab+b^2)$ or $(a-b)(a^2+ab+b^2)$.
3. Write the expression as a product of the two binomials.

Conclusion

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we provided a Q&A guide to help you understand the concept of factoring quadratic expressions. We discussed the different methods of factoring quadratic expressions, including factoring by grouping, factoring by using the greatest common factor (GCF), factoring by using the difference of squares, and factoring by using the sum and difference of cubes. By following the steps outlined in this article, you can master the art of factoring quadratic expressions and solve a wide range of algebraic problems.

Additional Resources

• Khan Academy: Factoring Quadratic Expressions
• Mathway: Factoring Quadratic Expressions
• Wolfram Alpha: Factoring Quadratic Expressions

Practice Problems

1. Factor the expression $x^2 + 5x + 6$ using the method of factoring by grouping.
2. Factor the expression $x^2 - 7x - 18$ using the method of factoring by using the greatest common factor (GCF).
3. Factor the expression $x^2 - 49$ using the method of factoring by using the difference of squares.
4. Factor the expression $x^3 + 27$ using the method of factoring by using the sum and difference of cubes.

Answer Key

1. $(x + 3)(x + 2)$
2. $(x - 9)(x + 2)$
3. $(x - 7)(x + 7)$
4. $(x + 3)(x^2 - 3x + 9)$