8z³-(x-y)³-(y+z)³-(z-x)³ Factorised​

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Introduction

In mathematics, factorization is a fundamental concept that involves breaking down complex expressions into simpler ones. It is a crucial skill for solving equations, simplifying expressions, and understanding various mathematical concepts. In this article, we will delve into the factorization of the expression 8z³-(x-y)³-(y+z)³-(z-x)³. We will explore the steps involved in factorizing this expression and provide a detailed explanation of the process.

Understanding the Expression

The given expression is 8z³-(x-y)³-(y+z)³-(z-x)³. At first glance, this expression may seem complex and daunting. However, by breaking it down into smaller components, we can simplify it and factorize it. Let's start by understanding the individual components of the expression.

  • 8z³: This is a simple cubic expression where 8 is the coefficient and z is the variable.
  • (x-y)³: This is a cubic expression in terms of x and y, where the difference between x and y is cubed.
  • (y+z)³: This is another cubic expression in terms of y and z, where the sum of y and z is cubed.
  • (z-x)³: This is a cubic expression in terms of z and x, where the difference between z and x is cubed.

Factorizing the Expression

To factorize the given expression, we can start by grouping the terms that have common factors. Let's group the first two terms and the last two terms separately.

8z³ - (x-y)³ - (y+z)³ - (z-x)³

= (8z³ - (x-y)³) - (y+z)³ - (z-x)³

Now, let's focus on the first group, 8z³ - (x-y)³. We can factor out a common factor of (x-y) from the second term.

8z³ - (x-y)³

= 8z³ - (x-y)(x-y)(x-y)

= 8z³ - (x-y)³

However, we can rewrite the first term as 8z³ = (x-y)(x-y)(x-y)(2z). This allows us to factor out a common factor of (x-y) from both terms.

8z³ - (x-y)³

= (x-y)(x-y)(x-y)(2z) - (x-y)(x-y)(x-y)

= (x-y)(x-y)(x-y)(2z - 1)

Now, let's focus on the second group, (y+z)³ - (z-x)³. We can factor out a common factor of (z-x) from the second term.

(y+z)³ - (z-x)³

= (y+z)³ - (z-x)(z-x)(z-x)

= (y+z)³ - (z-x)³

However, we can rewrite the first term as (y+z)³ = (z-x)(z-x)(z-x)(y+z). This allows us to factor out a common factor of (z-x) from both terms.

(y+z)³ - (z-x)³

= (z-x)(z-x)(z-x)(y+z) - (z-x)(z-x)(z-x)

= (z-x)(z-x)(z-x)(y+z - 1)

Combining the Groups

Now that we have factorized the two groups, we can combine them to get the final factorized expression.

(x-y)(x-y)(x-y)(2z - 1) - (z-x)(z-x)(z-x)(y+z - 1)

However, we can simplify this expression further by combining the two groups.

(x-y)(x-y)(x-y)(2z - 1) - (z-x)(z-x)(z-x)(y+z - 1)

= (x-y)(x-y)(x-y)(2z - 1) + (z-x)(z-x)(z-x)(1 - (y+z))

= (x-y)(x-y)(x-y)(2z - 1) + (z-x)(z-x)(z-x)(1 - (x+y))

Final Factorized Expression

After simplifying the expression, we get the final factorized form.

(x-y)(x-y)(x-y)(2z - 1) + (z-x)(z-x)(z-x)(1 - (x+y))

However, we can rewrite this expression in a more compact form.

(x-y)(x-y)(x-y)(2z - 1) + (z-x)(z-x)(z-x)(1 - x - y)

= (x-y)(x-y)(x-y)(2z - 1) + (z-x)(z-x)(z-x)(-x - y + 1)

= (x-y)(x-y)(x-y)(2z - 1) - (z-x)(z-x)(z-x)(x + y - 1)

Conclusion

In this article, we factorized the expression 8z³-(x-y)³-(y+z)³-(z-x)³. We broke down the expression into smaller components, grouped the terms, and factorized each group separately. Finally, we combined the groups to get the final factorized expression. The factorized form of the expression is (x-y)(x-y)(x-y)(2z - 1) - (z-x)(z-x)(z-x)(x + y - 1). This expression can be used to simplify complex equations and understand various mathematical concepts.

References

Introduction

In our previous article, we factorized the expression 8z³-(x-y)³-(y+z)³-(z-x)³. We broke down the expression into smaller components, grouped the terms, and factorized each group separately. In this article, we will answer some frequently asked questions related to the factorization of this expression.

Q&A

Q: What is the final factorized form of the expression 8z³-(x-y)³-(y+z)³-(z-x)³?

A: The final factorized form of the expression is (x-y)(x-y)(x-y)(2z - 1) - (z-x)(z-x)(z-x)(x + y - 1).

Q: How do I simplify complex expressions like 8z³-(x-y)³-(y+z)³-(z-x)³?

A: To simplify complex expressions, you can break them down into smaller components, group the terms, and factorize each group separately. This will help you to simplify the expression and understand the underlying mathematical concepts.

Q: What is the importance of factorization in mathematics?

A: Factorization is a fundamental concept in mathematics that involves breaking down complex expressions into simpler ones. It is a crucial skill for solving equations, simplifying expressions, and understanding various mathematical concepts.

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

A: To identify the common factors in an expression, you can look for terms that have common factors. You can then group these terms together and factorize them separately.

Q: What is the difference between a factor and a term?

A: A factor is a single component of an expression, while a term is a combination of factors. For example, in the expression 2x, 2 is a factor and x is a term.

Q: How do I rewrite an expression in a more compact form?

A: To rewrite an expression in a more compact form, you can combine like terms and simplify the expression. This will help you to make the expression more manageable and easier to understand.

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

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

  • Not identifying the common factors in an expression
  • Not grouping the terms correctly
  • Not simplifying the expression after factorizing
  • Not checking for errors in the factorization process

Conclusion

In this article, we answered some frequently asked questions related to the factorization of the expression 8z³-(x-y)³-(y+z)³-(z-x)³. We provided explanations and examples to help you understand the concepts and techniques involved in factorizing expressions. By following the steps outlined in this article, you can simplify complex expressions and understand various mathematical concepts.

References