What Is The Factorization Of $121 B^4 - 49$?A. $(11 B - 7)(11 B - 7)$B. $ ( 11 B + 7 ) ( 11 B − 7 ) (11 B + 7)(11 B - 7) ( 11 B + 7 ) ( 11 B − 7 ) [/tex]C. $\left(11 B^2 - 7\right)\left(11 B^2 - 7\right)$D. $\left(11 B^2 + 7\right)\left(11 B^2 -
Understanding the Problem
The given problem involves factorizing the expression $121 b^4 - 49$. To solve this, we need to apply the concept of difference of squares, which is a fundamental technique in algebra. The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$.
Breaking Down the Expression
We can rewrite the given expression as $121 b^4 - 49 = (11 b2)2 - 7^2$. This allows us to apply the difference of squares formula, where $a = 11 b^2$ and $b = 7$.
Applying the Difference of Squares Formula
Using the difference of squares formula, we can factorize the expression as follows:
Evaluating the Options
Now that we have factorized the expression, we can evaluate the given options:
A. $(11 b - 7)(11 b - 7)$
B. $(11 b + 7)(11 b - 7)$
C. $\left(11 b^2 - 7\right)\left(11 b^2 - 7\right)$
D. $\left(11 b^2 + 7\right)\left(11 b^2 - 7\right)$
Conclusion
Based on our factorization, we can see that the correct answer is:
D. $\left(11 b^2 + 7\right)\left(11 b^2 - 7\right)$
This is because our factorization matches the form of option D.
Why is this Important?
Factorizing expressions is an essential skill in mathematics, and it has numerous applications in various fields, such as physics, engineering, and computer science. By understanding how to factorize expressions, we can solve complex problems and make predictions about real-world phenomena.
Real-World Applications
Factorizing expressions has numerous real-world applications, including:
- Physics: Factorizing expressions is used to solve problems involving motion, energy, and momentum.
- Engineering: Factorizing expressions is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Computer Science: Factorizing expressions is used in algorithms and data structures, such as in the analysis of algorithms and the design of data structures.
Tips and Tricks
Here are some tips and tricks to help you factorize expressions:
- Use the difference of squares formula: The difference of squares formula is a powerful tool for factorizing expressions.
- Look for common factors: Look for common factors in the expression, such as a common term or a common factor.
- Use algebraic manipulations: Use algebraic manipulations, such as adding or subtracting terms, to simplify the expression.
Conclusion
In conclusion, factorizing expressions is an essential skill in mathematics, and it has numerous applications in various fields. By understanding how to factorize expressions, we can solve complex problems and make predictions about real-world phenomena.
Understanding the Problem
The given problem involves factorizing the expression $121 b^4 - 49$. To solve this, we need to apply the concept of difference of squares, which is a fundamental technique in algebra. The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$.
Q&A Session
Q: What is the difference of squares formula?
A: The difference of squares formula is a fundamental technique in algebra that states that $a^2 - b^2 = (a + b)(a - b)$.
Q: How do I apply the difference of squares formula?
A: To apply the difference of squares formula, you need to identify the two terms that are being subtracted, and then factorize them using the formula.
Q: What is the correct factorization of $121 b^4 - 49$?
A: The correct factorization of $121 b^4 - 49$ is $(11 b^2 + 7)(11 b^2 - 7)$.
Q: Why is this factorization correct?
A: This factorization is correct because it matches the form of the difference of squares formula, where $a = 11 b^2$ and $b = 7$.
Q: What are some real-world applications of factorizing expressions?
A: Factorizing expressions has numerous real-world applications, including physics, engineering, and computer science.
Q: How can I simplify complex expressions using algebraic manipulations?
A: You can simplify complex expressions using algebraic manipulations, such as adding or subtracting terms, to make them easier to factorize.
Q: What are some tips and tricks for factorizing expressions?
A: Some tips and tricks for factorizing expressions include using the difference of squares formula, looking for common factors, and using algebraic manipulations.
Conclusion
In conclusion, factorizing expressions is an essential skill in mathematics, and it has numerous applications in various fields. By understanding how to factorize expressions, we can solve complex problems and make predictions about real-world phenomena.
Frequently Asked Questions
Q: What is the difference of squares formula?
A: The difference of squares formula is a fundamental technique in algebra that states that $a^2 - b^2 = (a + b)(a - b)$.
Q: How do I apply the difference of squares formula?
A: To apply the difference of squares formula, you need to identify the two terms that are being subtracted, and then factorize them using the formula.
Q: What are some real-world applications of factorizing expressions?
A: Factorizing expressions has numerous real-world applications, including physics, engineering, and computer science.
Q: How can I simplify complex expressions using algebraic manipulations?
A: You can simplify complex expressions using algebraic manipulations, such as adding or subtracting terms, to make them easier to factorize.
Q: What are some tips and tricks for factorizing expressions?
A: Some tips and tricks for factorizing expressions include using the difference of squares formula, looking for common factors, and using algebraic manipulations.
Additional Resources
- Algebraic Manipulations: Learn how to simplify complex expressions using algebraic manipulations.
- Difference of Squares Formula: Learn how to apply the difference of squares formula to factorize expressions.
- Real-World Applications: Learn about the real-world applications of factorizing expressions in physics, engineering, and computer science.
Conclusion
In conclusion, factorizing expressions is an essential skill in mathematics, and it has numerous applications in various fields. By understanding how to factorize expressions, we can solve complex problems and make predictions about real-world phenomena.