What Is The Factored Form Of $36x^4 - 25$?$A. $(6x^2 - 5)(6x^2 + 5)$B. \$(6x^2 - 5)(6x^2 - 5)$[/tex\]C. $(6x^2 - 25)(6x^2 + 25)$D. $(6x^2 - 25)(6x^2 - 25)$

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What is the Factored Form of $36x^4 - 25$?

Understanding the Concept of 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 in mathematics, as it allows us to simplify complex expressions, solve equations, and analyze functions. In this article, we will explore the factored form of the expression $36x^4 - 25$ and examine the different options provided.

The Expression $36x^4 - 25$

The given expression is a difference of squares, which is a common algebraic expression that can be factored using the formula $(a^2 - b^2) = (a + b)(a - b)$. In this case, the expression can be written as $(6x2)2 - 5^2$.

Factoring the Expression

To factor the expression, we can use the difference of squares formula. We can rewrite the expression as $(6x2)2 - 5^2 = (6x^2 + 5)(6x^2 - 5)$. This is the factored form of the expression.

Evaluating the Options

Now that we have factored the expression, let's evaluate the options provided.

  • Option A: $(6x^2 - 5)(6x^2 + 5)$
  • Option B: $(6x^2 - 5)(6x^2 - 5)$
  • Option C: $(6x^2 - 25)(6x^2 + 25)$
  • Option D: $(6x^2 - 25)(6x^2 - 25)$

Analyzing the Options

Let's analyze each option and determine which one is correct.

  • Option A: $(6x^2 - 5)(6x^2 + 5)$ This option is correct, as it matches the factored form we obtained earlier.
  • Option B: $(6x^2 - 5)(6x^2 - 5)$ This option is incorrect, as it does not match the factored form we obtained earlier.
  • Option C: $(6x^2 - 25)(6x^2 + 25)$ This option is incorrect, as it does not match the factored form we obtained earlier.
  • Option D: $(6x^2 - 25)(6x^2 - 25)$ This option is incorrect, as it does not match the factored form we obtained earlier.

Conclusion

In conclusion, the factored form of $36x^4 - 25$ is $(6x^2 - 5)(6x^2 + 5)$. This is the correct option, and it matches the factored form we obtained earlier. Factoring is an essential skill in mathematics, and it allows us to simplify complex expressions and solve equations. By understanding the concept of factoring, we can analyze functions, solve problems, and make informed decisions.

Key Takeaways

  • Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions.
  • The expression $36x^4 - 25$ can be factored using the difference of squares formula.
  • The factored form of the expression is $(6x^2 - 5)(6x^2 + 5)$.
  • Factoring is an essential skill in mathematics that allows us to simplify complex expressions and solve equations.

Common Mistakes to Avoid

  • Not recognizing the difference of squares formula.
  • Not applying the formula correctly.
  • Not checking the options carefully.

Real-World Applications

  • Factoring is used in various real-world applications, such as cryptography, coding theory, and computer science.
  • Factoring is used to simplify complex expressions and solve equations in physics, engineering, and economics.
  • Factoring is used to analyze functions and make informed decisions in business and finance.

Final Thoughts

In conclusion, factoring is a crucial skill in mathematics that allows us to simplify complex expressions and solve equations. By understanding the concept of factoring, we can analyze functions, solve problems, and make informed decisions. The factored form of $36x^4 - 25$ is $(6x^2 - 5)(6x^2 + 5)$, and it is essential to recognize the difference of squares formula and apply it correctly.
Q&A: Factoring and Algebra

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about factoring and algebra.

Q: What is factoring?

A: Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. It is a crucial skill in mathematics that allows us to simplify complex expressions and solve equations.

Q: What is the difference of squares formula?

A: The difference of squares formula is a mathematical formula that states $(a^2 - b^2) = (a + b)(a - b)$. This formula is used to factor expressions that can be written in the form of a difference of squares.

Q: How do I factor an expression using the difference of squares formula?

A: To factor an expression using the difference of squares formula, you need to identify the two perfect squares that can be subtracted to obtain the original expression. Then, you can apply the formula to factor the expression.

Q: What is the factored form of $36x^4 - 25$?

A: The factored form of $36x^4 - 25$ is $(6x^2 - 5)(6x^2 + 5)$. This is obtained by applying the difference of squares formula to the expression.

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

A: Some common mistakes to avoid when factoring include not recognizing the difference of squares formula, not applying the formula correctly, and not checking the options carefully.

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

A: Factoring has various real-world applications, including cryptography, coding theory, and computer science. It is also used to simplify complex expressions and solve equations in physics, engineering, and economics.

Q: How can I practice factoring?

A: You can practice factoring by working on algebraic expressions and applying the difference of squares formula. You can also use online resources and practice problems to improve your skills.

Q: What are some tips for factoring?

A: Some tips for factoring include:

  • Recognizing the difference of squares formula
  • Applying the formula correctly
  • Checking the options carefully
  • Practicing regularly to improve your skills

Q: What are some common algebraic expressions that can be factored using the difference of squares formula?

A: Some common algebraic expressions that can be factored using the difference of squares formula include:

  • a2−b2a^2 - b^2

  • x2−y2x^2 - y^2

  • 36x4−2536x^4 - 25

Q: How can I use factoring to solve equations?

A: You can use factoring to solve equations by applying the difference of squares formula to the equation. This can help you simplify the equation and solve for the variable.

Q: What are some advanced topics in factoring?

A: Some advanced topics in factoring include:

  • Factoring quadratic expressions
  • Factoring polynomial expressions
  • Factoring rational expressions

Conclusion

In conclusion, factoring is a crucial skill in mathematics that allows us to simplify complex expressions and solve equations. By understanding the concept of factoring and applying the difference of squares formula, you can improve your skills and solve problems with ease. Remember to practice regularly and check your work carefully to avoid common mistakes.

Key Takeaways

  • Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions.
  • The difference of squares formula is a mathematical formula that states $(a^2 - b^2) = (a + b)(a - b)$.
  • Factoring has various real-world applications, including cryptography, coding theory, and computer science.
  • You can practice factoring by working on algebraic expressions and applying the difference of squares formula.

Final Thoughts

In conclusion, factoring is a crucial skill in mathematics that allows us to simplify complex expressions and solve equations. By understanding the concept of factoring and applying the difference of squares formula, you can improve your skills and solve problems with ease. Remember to practice regularly and check your work carefully to avoid common mistakes.