Factorizing the First Expression

Oct 13, 2025 by miodeya.com 68 views

Introduction

Factorizing expressions is a fundamental concept in algebra that involves breaking down complex expressions into simpler ones. It is a crucial skill that helps in solving equations, simplifying expressions, and understanding the underlying structure of mathematical problems. In this article, we will focus on factorizing two given expressions: $\frac{(x-1)^2}{(1-x)^2} = 1$ and $43a(a^2 + 4ab + 8b^2)$ . We will delve into the step-by-step process of factorizing these expressions and provide a comprehensive understanding of the underlying concepts.

Factorizing the First Expression

The first expression is $\frac{(x-1)^2}{(1-x)^2} = 1$ . To factorize this expression, we need to simplify it by canceling out common factors.

Step 1: Simplify the Expression

The given expression can be simplified by canceling out the common factors in the numerator and denominator.

$\frac{(x-1)^2}{(1-x)^2} = \frac{(x-1)(x-1)}{(1-x)(1-x)}$

Step 2: Factorize the Numerator and Denominator

Now, we can factorize the numerator and denominator separately.

$\frac{(x-1)(x-1)}{(1-x)(1-x)} = \frac{(x-1)^2}{(1-x)^2}$

Step 3: Cancel Out Common Factors

We can cancel out the common factors in the numerator and denominator.

$\frac{(x-1)^2}{(1-x)^2} = \frac{(x-1)(x-1)}{(1-x)(1-x)}$

Step 4: Simplify the Expression

After canceling out the common factors, we are left with a simplified expression.

$\frac{(x-1)(x-1)}{(1-x)(1-x)} = 1$

Conclusion

The factorized form of the first expression is $1$ . This means that the expression is already in its simplest form, and no further factorization is possible.

Factorizing the Second Expression

The second expression is $43a(a^2 + 4ab + 8b^2)$ . To factorize this expression, we need to identify the common factors and group them together.

Step 1: Identify Common Factors

The common factor in the expression is $a$ . We can factorize the expression by grouping the terms together.

$43a(a^2 + 4ab + 8b^2) = 43a(a(a + 4b) + 8b^2)$

Step 2: Factorize the Expression

Now, we can factorize the expression by grouping the terms together.

$43a(a(a + 4b) + 8b^2) = 43a(a(a + 4b) + 2^3b^2)$

Step 3: Factorize the Expression Further

We can factorize the expression further by grouping the terms together.

$43a(a(a + 4b) + 2^3b^2) = 43a(a(a + 2b) + 2b(a + 2b))$

Step 4: Factorize the Expression Completely

After grouping the terms together, we can factorize the expression completely.

$43a(a(a + 2b) + 2b(a + 2b)) = 43a(a + 2b)(a + 2b)$

Conclusion

The factorized form of the second expression is $43a(a + 2b)(a + 2b)$ . This means that the expression has been broken down into its simplest form, and no further factorization is possible.

Conclusion

$In conclusion, factorizing expressions is a crucial skill that helps in solving equations, simplifying expressions, and understanding the underlying structure of mathematical problems. We have factorized two given expressions: $\frac{(x-1)^2}{(1-x)^2} = 1$ and $43a(a^2 + 4ab + 8b^2)$. By following the step-by-step process of factorizing these expressions, we have provided a comprehensive understanding of the underlying concepts. We hope that this article has been informative and helpful in understanding the concept of factorizing expressions.<br/> **Factorizing Expressions: A Comprehensive Guide**$

Q&A: Frequently Asked Questions

Q: What is factorizing an expression?

A: Factorizing an expression is the process of breaking down a complex expression into simpler ones. It involves identifying the common factors and grouping them together to simplify the expression.

Q: Why is factorizing an expression important?

A: Factorizing an expression is important because it helps in solving equations, simplifying expressions, and understanding the underlying structure of mathematical problems. It also helps in identifying the roots of an equation and in solving systems of equations.

Q: How do I factorize an expression?

A: To factorize an expression, you need to identify the common factors and group them together. You can use various techniques such as grouping, factoring out the greatest common factor, and using the difference of squares formula.

Q: What are the common techniques used to factorize an expression?

A: The common techniques used to factorize an expression include:

Grouping: This involves grouping the terms of the expression together to identify the common factors.
Factoring out the greatest common factor: This involves factoring out the greatest common factor from the terms of the expression.
Using the difference of squares formula: This involves using the formula $a^2 - b^2 = (a + b)(a - b)$ to factorize the expression.

Q: How do I identify the common factors in an expression?

A: To identify the common factors in an expression, you need to look for the terms that have a common factor. You can use the distributive property to expand the expression and identify the common factors.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . This property can be used to expand an expression and identify the common factors.

Q: How do I use the distributive property to factorize an expression?

A: To use the distributive property to factorize an expression, you need to expand the expression and identify the common factors. You can then group the terms together to simplify the expression.

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

A: Some common mistakes to avoid when factorizing an expression include:

Not identifying the common factors correctly
Not grouping the terms correctly
Not using the distributive property correctly
Not checking the expression for any errors

Q: How do I check an expression for errors?

A: To check an expression for errors, you need to plug in some values for the variables and see if the expression holds true. You can also use a calculator to check the expression.

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

A: Some real-world applications of factorizing expressions include:

Solving systems of equations
Finding the roots of an equation
Simplifying complex expressions
Solving optimization problems

Conclusion

In conclusion, factorizing expressions is a crucial skill that helps in solving equations, simplifying expressions, and understanding the underlying structure of mathematical problems. By following the step-by-step process of factorizing an expression, you can identify the common factors and group them together to simplify the expression. We hope that this article has been informative and helpful in understanding the concept of factorizing expressions.