What Is The Factored Form Of $n^2 - 25$?A. $(n-25)(n-1)$ B. $ ( N − 5 ) ( N + 5 ) (n-5)(n+5) ( N − 5 ) ( N + 5 ) [/tex] C. $(n+5)(n+5)$ D. $(n-25)(n+1)$

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Introduction to Factoring

Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. It is a crucial skill for solving equations, simplifying expressions, and understanding the properties of functions. In this article, we will explore the factored form of the expression $n^2 - 25$ and examine the different options provided.

Understanding the Expression

The given expression is $n^2 - 25$. This is a quadratic expression, which means it is a polynomial of degree two. The expression can be rewritten as $n^2 - 5^2$, using the difference of squares formula. This formula states that $a^2 - b^2 = (a + b)(a - b)$.

Factoring the Expression

Using the difference of squares formula, we can factor the expression $n^2 - 25$ as follows:

n2−25=(n+5)(n−5)n^2 - 25 = (n + 5)(n - 5)

This is the factored form of the expression. It is a product of two binomials, each of which is a linear expression.

Examining the Options

Now that we have found the factored form of the expression, let's examine the options provided:

A. $(n-25)(n-1)$ B. $(n-5)(n+5)$ C. $(n+5)(n+5)$ D. $(n-25)(n+1)$

Option A: $(n-25)(n-1)$

This option is incorrect because it does not match the factored form we found. The first term in the product is $(n-25)$, which is not equal to $(n+5)$.

Option B: $(n-5)(n+5)$

This option is correct because it matches the factored form we found. The product of $(n-5)$ and $(n+5)$ is indeed $(n^2 - 25)$.

Option C: $(n+5)(n+5)$

This option is incorrect because it does not match the factored form we found. The product of $(n+5)$ and $(n+5)$ is $(n^2 + 10n + 25)$, which is not equal to $(n^2 - 25)$.

Option D: $(n-25)(n+1)$

This option is incorrect because it does not match the factored form we found. The product of $(n-25)$ and $(n+1)$ is $(n^2 - 24n - 25)$, which is not equal to $(n^2 - 25)$.

Conclusion

In conclusion, the factored form of $n^2 - 25$ is $(n-5)(n+5)$. This is the correct answer among the options provided. Factoring is an essential skill in algebra, and understanding how to factor expressions is crucial for solving equations and simplifying expressions.

Tips for Factoring

Here are some tips for factoring expressions:

  • Use the difference of squares formula to factor expressions of the form $a^2 - b^2$.
  • Look for common factors in the terms of the expression.
  • Use the distributive property to expand the expression and identify the factors.
  • Use the factored form to simplify the expression and solve equations.

Practice Problems

Here are some practice problems to help you practice factoring expressions:

  • Factor the expression $x^2 - 9$.
  • Factor the expression $y^2 - 16$.
  • Factor the expression $z^2 - 25$.

Solutions to Practice Problems

Here are the solutions to the practice problems:

  • x2−9=(x+3)(x−3)x^2 - 9 = (x + 3)(x - 3)

  • y2−16=(y+4)(y−4)y^2 - 16 = (y + 4)(y - 4)

  • z2−25=(z+5)(z−5)z^2 - 25 = (z + 5)(z - 5)

Conclusion

In conclusion, factoring is an essential skill in algebra that involves expressing an algebraic expression as a product of simpler expressions. The factored form of $n^2 - 25$ is $(n-5)(n+5)$. Understanding how to factor expressions is crucial for solving equations and simplifying expressions. With practice and patience, you can become proficient in factoring expressions and solve a wide range of problems in algebra.

Introduction

Factoring expressions is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. It is a crucial skill for solving equations, simplifying expressions, and understanding the properties of functions. In this article, we will provide a Q&A guide to help you understand the concept of factoring expressions and how to apply it to solve problems.

Q: What is factoring?

A: Factoring is the process of expressing an algebraic expression as a product of simpler expressions. It involves breaking down a complex expression into smaller, more manageable parts.

Q: Why is factoring important?

A: Factoring is important because it allows us to simplify complex expressions, solve equations, and understand the properties of functions. It is a crucial skill for solving problems in algebra and other areas of mathematics.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

  • Difference of squares: This involves factoring expressions of the form $a^2 - b^2$.
  • Common factors: This involves factoring expressions that have common factors in the terms.
  • Distributive property: This involves using the distributive property to expand the expression and identify the factors.

Q: How do I factor an expression?

A: To factor an expression, follow these steps:

  1. Look for common factors: Identify any common factors in the terms of the expression.
  2. Use the difference of squares formula: If the expression is of the form $a^2 - b^2$, use the difference of squares formula to factor it.
  3. Use the distributive property: Use the distributive property to expand the expression and identify the factors.
  4. Simplify the expression: Simplify the expression by combining like terms.

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

A: Some common mistakes to avoid when factoring include:

  • Not identifying common factors: Failing to identify common factors in the terms of the expression.
  • Not using the difference of squares formula: Failing to use the difference of squares formula when the expression is of the form $a^2 - b^2$.
  • Not simplifying the expression: Failing to simplify the expression by combining like terms.

Q: How do I practice factoring?

A: To practice factoring, try the following:

  • Work through examples: Work through examples of factoring expressions to practice the skill.
  • Use online resources: Use online resources, such as factoring worksheets and practice problems, to practice factoring.
  • Take practice tests: Take practice tests to assess your understanding of factoring and identify areas for improvement.

Q: What are some real-world applications of factoring?

A: Factoring has many real-world applications, including:

  • Simplifying complex expressions: Factoring can be used to simplify complex expressions in fields such as engineering and physics.
  • Solving equations: Factoring can be used to solve equations in fields such as economics and finance.
  • Understanding functions: Factoring can be used to understand the properties of functions in fields such as computer science and data analysis.

Conclusion

In conclusion, factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. It is a crucial skill for solving equations, simplifying expressions, and understanding the properties of functions. By following the steps outlined in this article and practicing factoring, you can become proficient in this skill and apply it to solve a wide range of problems in algebra and other areas of mathematics.

Additional Resources

Here are some additional resources to help you practice factoring:

  • Factoring worksheets: Download factoring worksheets to practice factoring expressions.
  • Practice problems: Work through practice problems to assess your understanding of factoring and identify areas for improvement.
  • Online resources: Use online resources, such as factoring tutorials and videos, to learn more about factoring and how to apply it to solve problems.

Final Tips

Here are some final tips to help you master factoring:

  • Practice regularly: Practice factoring regularly to develop your skills and build your confidence.
  • Use online resources: Use online resources, such as factoring worksheets and practice problems, to practice factoring and assess your understanding.
  • Seek help when needed: Don't be afraid to seek help when you need it. Ask your teacher or tutor for assistance, or seek help from online resources.