Factorize $25x^2 - 4$.A. $(5x - 2)(5x - 2)$B. $ ( 5 X − 2 ) ( 5 X + 2 ) (5x - 2)(5x + 2) ( 5 X − 2 ) ( 5 X + 2 ) [/tex]C. $(5x - 2)(5 + 2x)$D. $(5x - 2)(5 - 2x)$

$$

Introduction

In algebra, factorization is a process of expressing an algebraic expression as a product of simpler expressions. It is an essential concept in mathematics, and it plays a crucial role in solving equations and inequalities. In this article, we will focus on factorizing the expression $25x^2 - 4$. We will explore the different methods of factorization and provide step-by-step solutions to help you understand the concept better.

What is Factorization?

Factorization is the process of expressing an algebraic expression as a product of simpler expressions. It involves breaking down a complex expression into smaller parts, called factors, which can be multiplied together to obtain the original expression. Factorization is a powerful tool in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics.

Methods of Factorization

There are several methods of factorization, including:

• Factoring out the greatest common factor (GCF): This method involves factoring out the greatest common factor from each term in the expression.
• Factoring by grouping: This method involves grouping the terms in the expression into pairs and factoring out the common factors from each pair.
• Factoring quadratic expressions: This method involves factoring quadratic expressions in the form of $ax^2 + bx + c$.

Factorizing $25x^2 - 4$

To factorize the expression $25x^2 - 4$, we can use the method of factoring quadratic expressions. We can start by recognizing that $25x^2 - 4$ is a difference of squares, which can be factored as $(a^2 - b^2) = (a + b)(a - b)$.

Step 1: Identify the difference of squares

The expression $25x^2 - 4$ can be written as $(5x)^2 - 2^2$, which is a difference of squares.

Step 2: Factor the difference of squares

Using the formula for factoring a difference of squares, we can write:

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

Step 3: Simplify the expression

The expression $(5x + 2)(5x - 2)$ is already in its simplest form.

Conclusion

In conclusion, the correct factorization of the expression $25x^2 - 4$ is $(5x + 2)(5x - 2)$. This can be verified by multiplying the two factors together, which will give us the original expression.

Comparison with Other Options

Let's compare our answer with the other options provided:

• Option A: $(5x - 2)(5x - 2)$ is not correct because it is a repeated factor, and it does not match the original expression.
• Option B: $(5x - 2)(5x + 2)$ is not correct because it does not match the original expression.
• Option C: $(5x - 2)(5 + 2x)$ is not correct because it does not match the original expression.
• Option D: $(5x - 2)(5 - 2x)$ is not correct because it does not match the original expression.

Final Answer

The correct factorization of the expression $25x^2 - 4$ is $(5x + 2)(5x - 2)$.

Frequently Asked Questions

• Q: What is factorization? A: Factorization is the process of expressing an algebraic expression as a product of simpler expressions.
• Q: What are the different methods of factorization? A: There are several methods of factorization, including factoring out the greatest common factor (GCF), factoring by grouping, and factoring quadratic expressions.
• Q: How do I factorize a quadratic expression? A: To factorize a quadratic expression, you can use the method of factoring quadratic expressions, which involves recognizing the expression as a difference of squares and factoring it accordingly.

Conclusion

In conclusion, factorization is a powerful tool in mathematics that involves expressing an algebraic expression as a product of simpler expressions. In this article, we have explored the different methods of factorization and provided step-by-step solutions to help you understand the concept better. We have also compared our answer with the other options provided and concluded that the correct factorization of the expression $25x^2 - 4$ is $(5x + 2)(5x - 2)$.

$$

Introduction

In our previous article, we explored the concept of factorization and provided a step-by-step solution to factorize the expression $25x^2 - 4$. In this article, we will answer some of the frequently asked questions related to factorization and provide additional insights to help you understand the concept better.

Q&A

Q: What is factorization?

A: Factorization is the process of expressing an algebraic expression as a product of simpler expressions. It involves breaking down a complex expression into smaller parts, called factors, which can be multiplied together to obtain the original expression.

Q: What are the different methods of factorization?

A: There are several methods of factorization, including:

• Factoring out the greatest common factor (GCF): This method involves factoring out the greatest common factor from each term in the expression.
• Factoring by grouping: This method involves grouping the terms in the expression into pairs and factoring out the common factors from each pair.
• Factoring quadratic expressions: This method involves factoring quadratic expressions in the form of $ax^2 + bx + c$.

Q: How do I factorize a quadratic expression?

A: To factorize a quadratic expression, you can use the method of factoring quadratic expressions, which involves recognizing the expression as a difference of squares and factoring it accordingly.

Q: What is a difference of squares?

A: A difference of squares is an expression of the form $a^2 - b^2$, which can be factored as $(a + b)(a - b)$.

Q: How do I factor a difference of squares?

A: To factor a difference of squares, you can use the formula $(a + b)(a - b)$, where $a$ and $b$ are the two terms in the expression.

Q: What is the correct factorization of the expression $25x^2 - 4$?

A: The correct factorization of the expression $25x^2 - 4$ is $(5x + 2)(5x - 2)$.

Q: Why is option A not correct?

A: Option A is not correct because it is a repeated factor, and it does not match the original expression.

Q: Why is option B not correct?

A: Option B is not correct because it does not match the original expression.

Q: Why is option C not correct?

A: Option C is not correct because it does not match the original expression.

Q: Why is option D not correct?

A: Option D is not correct because it does not match the original expression.

Additional Insights

• Factorization is a powerful tool in mathematics that involves expressing an algebraic expression as a product of simpler expressions.
• There are several methods of factorization, including factoring out the greatest common factor (GCF), factoring by grouping, and factoring quadratic expressions.
• To factorize a quadratic expression, you can use the method of factoring quadratic expressions, which involves recognizing the expression as a difference of squares and factoring it accordingly.
• A difference of squares is an expression of the form $a^2 - b^2$, which can be factored as $(a + b)(a - b)$.

Conclusion

In conclusion, factorization is a powerful tool in mathematics that involves expressing an algebraic expression as a product of simpler expressions. In this article, we have answered some of the frequently asked questions related to factorization and provided additional insights to help you understand the concept better. We have also compared our answer with the other options provided and concluded that the correct factorization of the expression $25x^2 - 4$ is $(5x + 2)(5x - 2)$.