Find The Value Of X Y ( X + Y ) + Y Z ( Y + Z ) + Z X ( Z + X ) + 3 X Y Z X Y(x+y)+y Z(y+z)+z X(z+x)+3 X Y Z X Y ( X + Y ) + Yz ( Y + Z ) + Z X ( Z + X ) + 3 X Yz When X = A ( B − C ) , Y = B ( C − A ) , Z = C ( A − B X=a(b-c), Y=b(c-a), Z=c(a-b X = A ( B − C ) , Y = B ( C − A ) , Z = C ( A − B ].

Introduction

In this article, we will delve into the world of algebra and explore a complex expression involving variables $x$, $y$, and $z$. The expression is given as $x y(x+y)+y z(y+z)+z x(z+x)+3 x y z$, and we are asked to find its value when $x=a(b-c), y=b(c-a), z=c(a-b)$. This problem requires us to substitute the given values of $x$, $y$, and $z$ into the expression and simplify it to obtain the final result.

Understanding the Given Values

Before we proceed with the substitution, let's take a closer look at the given values of $x$, $y$, and $z$. We are given that:

$x=a(b-c)$ $y=b(c-a)$ $z=c(a-b)$

These values represent the relationships between the variables $a$, $b$, and $c$. We can see that each variable is expressed in terms of the other two variables, which will be essential in simplifying the expression.

Substituting the Values into the Expression

Now that we have a clear understanding of the given values, let's substitute them into the expression $x y(x+y)+y z(y+z)+z x(z+x)+3 x y z$. We will replace each variable with its corresponding value and simplify the expression step by step.

$x y(x+y)+y z(y+z)+z x(z+x)+3 x y z$ $= a(b-c) \cdot b(c-a) (a(b-c)+b(c-a))+b(c-a) \cdot c(a-b) (b(c-a)+c(a-b))+c(a-b) \cdot a(b-c) (c(a-b)+a(b-c))+3 a(b-c) \cdot b(c-a) \cdot c(a-b)$

Simplifying the Expression

Now that we have substituted the values into the expression, let's simplify it step by step. We will start by expanding the products and then combine like terms.

$= a(b-c) \cdot b(c-a) (ab-ac+bc-ab)+b(c-a) \cdot c(a-b) (bc-ba+ca-ab)+c(a-b) \cdot a(b-c) (ca-cb+ab-bc)+3 a(b-c) \cdot b(c-a) \cdot c(a-b)$

Expanding the Products

Let's expand the products in the expression and simplify the terms.

$= a(b-c) \cdot b(c-a) (ab-ac+bc-ab)+b(c-a) \cdot c(a-b) (bc-ba+ca-ab)+c(a-b) \cdot a(b-c) (ca-cb+ab-bc)+3 a(b-c) \cdot b(c-a) \cdot c(a-b)$ $= a(b-c) \cdot b(c-a) (2bc-2ab)+b(c-a) \cdot c(a-b) (2ca-2ab)+c(a-b) \cdot a(b-c) (2ab-2bc)+3 a(b-c) \cdot b(c-a) \cdot c(a-b)$

Combining Like Terms

Now that we have expanded the products, let's combine like terms in the expression.

$= a(b-c) \cdot b(c-a) (2bc-2ab)+b(c-a) \cdot c(a-b) (2ca-2ab)+c(a-b) \cdot a(b-c) (2ab-2bc)+3 a(b-c) \cdot b(c-a) \cdot c(a-b)$ $= 2abc(b-c)-2ab^2(c-a)+2bca(c-a)-2abc(a-b)+3 a(b-c) \cdot b(c-a) \cdot c(a-b)$

Simplifying the Terms

Let's simplify the terms in the expression by combining like terms.

$= 2abc(b-c)-2ab^2(c-a)+2bca(c-a)-2abc(a-b)+3 a(b-c) \cdot b(c-a) \cdot c(a-b)$ $= 2abc(b-c)-2ab^2(c-a)+2bca(c-a)-2abc(a-b)+3 a^2 b^2 c^2$

Final Result

After simplifying the expression, we can see that the final result is:

$= 3 a^2 b^2 c^2$

This is the value of the complex algebraic expression when $x=a(b-c), y=b(c-a), z=c(a-b)$.

Conclusion

In this article, we have explored a complex algebraic expression involving variables $x$, $y$, and $z$. We have substituted the given values of $x$, $y$, and $z$ into the expression and simplified it to obtain the final result. The final result is $3 a^2 b^2 c^2$, which represents the value of the complex expression when $x=a(b-c), y=b(c-a), z=c(a-b)$. This problem requires a deep understanding of algebraic expressions and the ability to simplify complex terms.

Introduction

In our previous article, we explored a complex algebraic expression involving variables $x$, $y$, and $z$. We substituted the given values of $x$, $y$, and $z$ into the expression and simplified it to obtain the final result. In this article, we will answer some frequently asked questions (FAQs) about the complex algebraic expression.

Q: What is the value of the complex algebraic expression?

A: The value of the complex algebraic expression is $3 a^2 b^2 c^2$.

Q: How do I simplify the complex algebraic expression?

A: To simplify the complex algebraic expression, you need to substitute the given values of $x$, $y$, and $z$ into the expression and then simplify the terms by combining like terms.

Q: What is the relationship between the variables $a$, $b$, and $c$?

A: The variables $a$, $b$, and $c$ are related to each other through the given values of $x$, $y$, and $z$. Specifically, $x=a(b-c)$, $y=b(c-a)$, and $z=c(a-b)$.

Q: How do I expand the products in the expression?

A: To expand the products in the expression, you need to multiply the terms together and then simplify the resulting expression.

Q: What is the final result of the complex algebraic expression?

A: The final result of the complex algebraic expression is $3 a^2 b^2 c^2$.

Q: Can I use this method to simplify other complex algebraic expressions?

A: Yes, you can use this method to simplify other complex algebraic expressions by substituting the given values of the variables into the expression and then simplifying the terms by combining like terms.

Q: What are some common mistakes to avoid when simplifying complex algebraic expressions?

A: Some common mistakes to avoid when simplifying complex algebraic expressions include:

• Not substituting the given values of the variables into the expression
• Not simplifying the terms by combining like terms
• Not expanding the products in the expression
• Not checking the final result for errors

Q: How can I practice simplifying complex algebraic expressions?

A: You can practice simplifying complex algebraic expressions by working through examples and exercises in your textbook or online resources. You can also try simplifying your own complex algebraic expressions to practice your skills.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about the complex algebraic expression. We have provided step-by-step instructions on how to simplify the expression and have highlighted some common mistakes to avoid. We hope that this article has been helpful in clarifying any questions you may have had about the complex algebraic expression.

Additional Resources

If you are looking for additional resources to help you simplify complex algebraic expressions, here are a few suggestions:

• Online resources: There are many online resources available that can help you simplify complex algebraic expressions, including video tutorials, interactive exercises, and online calculators.
• Textbooks: Your textbook may have a section on simplifying complex algebraic expressions that you can refer to for help.
• Practice problems: Working through practice problems can help you develop your skills in simplifying complex algebraic expressions.
• Online communities: Joining online communities, such as forums or social media groups, can connect you with other students who are also learning to simplify complex algebraic expressions.

Final Thoughts

Simplifying complex algebraic expressions can be a challenging task, but with practice and patience, you can develop the skills you need to succeed. Remember to always substitute the given values of the variables into the expression, simplify the terms by combining like terms, and expand the products in the expression. With these skills, you will be able to simplify complex algebraic expressions with ease.