Factorize The Following Expressions:1. $C^2 + D^2$2. $\frac{C^2}{25} - \frac{f^2}{36}$

Introduction

Factorizing algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions. This technique is essential in solving equations, simplifying expressions, and understanding the properties of functions. In this article, we will focus on factorizing two given expressions: $C^2 + d^2$ and $\frac{C^2}{25} - \frac{f^2}{36}$.

Expression 1: $C^2 + d^2$

The first expression we will factorize is $C^2 + d^2$. This expression represents the sum of two squares, which can be factorized using the following formula:

$$a^2 + b^2 = (a + b)^2 - 2ab$$

However, in this case, we can use a more straightforward approach by recognizing that $C^2 + d^2$ is a sum of two squares. We can rewrite the expression as:

$$C^2 + d^2 = (C + d)^2 - 2Cd$$

But, we can also factorize it using the difference of squares formula, which is not applicable here. However, we can use the factoring by grouping method.

$$C^2 + d^2 = (C + d)(C - d)$$

This is the most simplified form of the expression.

Expression 2: $\frac{C^2}{25} - \frac{f^2}{36}$

The second expression we will factorize is $\frac{C^2}{25} - \frac{f^2}{36}$. This expression represents the difference of two squares, which can be factorized using the following formula:

$$a^2 - b^2 = (a + b)(a - b)$$

However, in this case, we have fractions, so we need to find a common denominator to combine the fractions. The least common multiple of 25 and 36 is 900.

$$\frac{C^2}{25} - \frac{f^2}{36} = \frac{36C^2}{900} - \frac{25f^2}{900}$$

Now, we can factorize the expression using the difference of squares formula:

$$\frac{36C^2}{900} - \frac{25f^2}{900} = \frac{(6C - 5f)(6C + 5f)}{900}$$

This is the most simplified form of the expression.

Conclusion

In this article, we factorized two algebraic expressions: $C^2 + d^2$ and $\frac{C^2}{25} - \frac{f^2}{36}$. We used various techniques, including the difference of squares formula and factoring by grouping, to simplify the expressions. The factorized forms of the expressions are $(C + d)(C - d)$ and $\frac{(6C - 5f)(6C + 5f)}{900}$, respectively.

Tips and Tricks

• When factorizing expressions, always look for common factors or patterns.
• Use the difference of squares formula to factorize expressions of the form $a^2 - b^2$.
• When working with fractions, find a common denominator to combine the fractions.
• Use factoring by grouping to factorize expressions that do not fit the difference of squares formula.

Practice Problems

1. Factorize the expression $x^2 + 4x + 4$.
2. Factorize the expression $\frac{x^2}{9} - \frac{y^2}{16}$.
3. Factorize the expression $a^2 - 2ab + b^2$.

Solutions

1. $x^2 + 4x + 4 = (x + 2)^2$
2. $\frac{x^2}{9} - \frac{y^2}{16} = \frac{(x - 2y)(x + 2y)}{72}$
3. $a^2 - 2ab + b^2 = (a - b)^2$

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Q&A: Factorizing Algebraic Expressions

Q: What is factorizing an algebraic expression?

A: Factorizing an algebraic expression involves expressing the expression as a product of simpler expressions. This technique is essential in solving equations, simplifying expressions, and understanding the properties of functions.

Q: What are the different types of factorization?

A: There are several types of factorization, including:

• Difference of squares: This involves factorizing expressions of the form $a^2 - b^2$.
• Sum of squares: This involves factorizing expressions of the form $a^2 + b^2$.
• Factoring by grouping: This involves factorizing expressions by grouping terms together.
• Common factorization: This involves factorizing expressions by identifying common factors.

Q: How do I factorize an expression?

A: To factorize an expression, follow these steps:

1. Identify the type of factorization: Determine whether the expression is a difference of squares, sum of squares, or another type of factorization.
2. Apply the factorization formula: Use the appropriate formula to factorize the expression.
3. Simplify the expression: Simplify the expression by combining like terms.

Q: What are some common mistakes to avoid when factorizing expressions?

A: Some common mistakes to avoid when factorizing expressions include:

• Not identifying the type of factorization: Failing to identify the type of factorization can lead to incorrect factorization.
• Not applying the correct formula: Applying the wrong formula can lead to incorrect factorization.
• Not simplifying the expression: Failing to simplify the expression can lead to incorrect solutions.

Q: How do I check my factorization?

A: To check your factorization, follow these steps:

1. Multiply the factors: Multiply the factors together to see if they equal the original expression.
2. Simplify the expression: Simplify the expression to see if it equals the original expression.
3. Verify the solution: Verify that the solution is correct by plugging it back into the original equation.

Q: What are some real-world applications of factorizing expressions?

A: Factorizing expressions has many real-world applications, including:

• Solving equations: Factorizing expressions is essential in solving equations, such as quadratic equations.
• Simplifying expressions: Factorizing expressions can simplify complex expressions, making them easier to work with.
• Understanding functions: Factorizing expressions can help understand the properties of functions, such as their domain and range.

Q: Can you provide some examples of factorizing expressions?

A: Here are some examples of factorizing expressions:

• Example 1: Factorize the expression $x^2 + 4x + 4$.
• Example 2: Factorize the expression $\frac{x^2}{9} - \frac{y^2}{16}$.
• Example 3: Factorize the expression $a^2 - 2ab + b^2$.

Solutions

• Example 1: $x^2 + 4x + 4 = (x + 2)^2$
• Example 2: $\frac{x^2}{9} - \frac{y^2}{16} = \frac{(x - 2y)(x + 2y)}{72}$
• Example 3: $a^2 - 2ab + b^2 = (a - b)^2$

Conclusion

In this article, we have discussed the concept of factorizing algebraic expressions, including the different types of factorization, how to factorize expressions, and common mistakes to avoid. We have also provided some examples of factorizing expressions and discussed the real-world applications of factorizing expressions.