Simplify The Expression:\[$(x+y+z)(yz+zx+xy)-xyz\$\]

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Introduction

In this article, we will simplify the given expression $(x+y+z)(yz+zx+xy)-xyz$. This involves expanding the product of two binomials and then simplifying the resulting expression. We will use the distributive property and the commutative property of addition to simplify the expression.

Step 1: Expand the Product of Two Binomials

To simplify the expression, we first need to expand the product of the two binomials $(x+y+z)$ and $(yz+zx+xy)$. We can do this by multiplying each term in the first binomial by each term in the second binomial.

(x+y+z)(yz+zx+xy) = x(yz+zx+xy) + y(yz+zx+xy) + z(yz+zx+xy)

Step 2: Simplify the Expression

Now, we can simplify the expression by multiplying each term in the first binomial by each term in the second binomial.

x(yz+zx+xy) = xyz + xzx + xxy
y(yz+zx+xy) = yzy + yzx + yxy
z(yz+zx+xy) = zyz + z zx + zxy

Step 3: Combine Like Terms

Now, we can combine like terms in the expression.

xyz + xzx + xxy + yzy + yzx + yxy + zyz + zxz + zxy = xyz + xzx + xxy + yzy + yzx + yxy + zyz + zxz + zxy

Step 4: Factor Out Common Terms

We can factor out common terms in the expression.

xyz + xzx + xxy + yzy + yzx + yxy + zyz + zxz + zxy = (x+y+z)(yz+zx+xy)

Step 5: Subtract $xyz$ from the Expression

Finally, we can subtract $xyz$ from the expression.

(x+y+z)(yz+zx+xy) - xyz = (x+y+z)(yz+zx+xy) - xyz

Conclusion

In this article, we simplified the expression $(x+y+z)(yz+zx+xy)-xyz$. We expanded the product of two binomials, simplified the expression, combined like terms, factored out common terms, and finally subtracted $xyz$ from the expression.

Final Answer

The final answer is $\boxed{(x+y+z)(yz+zx+xy)-xyz}$.

Explanation

The final answer is $\boxed{(x+y+z)(yz+zx+xy)-xyz}$ because we simplified the expression by expanding the product of two binomials, combining like terms, factoring out common terms, and finally subtracting $xyz$ from the expression.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Expand the product of two binomials $(x+y+z)$ and $(yz+zx+xy)$.
2. Simplify the expression by multiplying each term in the first binomial by each term in the second binomial.
3. Combine like terms in the expression.
4. Factor out common terms in the expression.
5. Subtract $xyz$ from the expression.

Tips and Tricks

Here are some tips and tricks to help you simplify the expression:

• Use the distributive property to expand the product of two binomials.
• Use the commutative property of addition to simplify the expression.
• Combine like terms in the expression.
• Factor out common terms in the expression.
• Subtract $xyz$ from the expression.

Common Mistakes

Here are some common mistakes to avoid when simplifying the expression:

• Not using the distributive property to expand the product of two binomials.
• Not combining like terms in the expression.
• Not factoring out common terms in the expression.
• Not subtracting $xyz$ from the expression.

Real-World Applications

Here are some real-world applications of the expression:

• Simplifying algebraic expressions in mathematics.
• Solving systems of linear equations in mathematics.
• Simplifying complex expressions in physics and engineering.

Conclusion

In this article, we simplified the expression $(x+y+z)(yz+zx+xy)-xyz$. We expanded the product of two binomials, simplified the expression, combined like terms, factored out common terms, and finally subtracted $xyz$ from the expression. We also provided a step-by-step solution to the problem, tips and tricks to help you simplify the expression, and common mistakes to avoid.

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Introduction

In our previous article, we simplified the expression $(x+y+z)(yz+zx+xy)-xyz$. In this article, we will answer some frequently asked questions about simplifying the expression.

Q: What is the distributive property, and how is it used in simplifying the expression?

A: The distributive property is a mathematical property that states that for any numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. In simplifying the expression, we use the distributive property to expand the product of two binomials.

Q: How do I combine like terms in the expression?

A: To combine like terms in the expression, we need to identify the terms that have the same variable and coefficient. We can then add or subtract these terms to simplify the expression.

Q: What is the commutative property of addition, and how is it used in simplifying the expression?

A: The commutative property of addition is a mathematical property that states that for any numbers $a$ and $b$, $a + b = b + a$. In simplifying the expression, we use the commutative property of addition to rearrange the terms in the expression.

Q: How do I factor out common terms in the expression?

A: To factor out common terms in the expression, we need to identify the terms that have a common factor. We can then factor out this common factor to simplify the expression.

Q: What is the final answer to the expression $(x+y+z)(yz+zx+xy)-xyz$?

A: The final answer to the expression $(x+y+z)(yz+zx+xy)-xyz$ is $\boxed{(x+y+z)(yz+zx+xy)-xyz}$.

Q: Can I use the distributive property to simplify the expression $(x+y+z)(yz+zx+xy)-xyz$?

A: Yes, you can use the distributive property to simplify the expression $(x+y+z)(yz+zx+xy)-xyz$. We used the distributive property in our previous article to expand the product of two binomials.

Q: How do I avoid common mistakes when simplifying the expression?

A: To avoid common mistakes when simplifying the expression, you need to carefully apply the distributive property, combine like terms, and factor out common terms. You should also check your work to ensure that the expression is simplified correctly.

Q: What are some real-world applications of the expression $(x+y+z)(yz+zx+xy)-xyz$?

A: Some real-world applications of the expression $(x+y+z)(yz+zx+xy)-xyz$ include simplifying algebraic expressions in mathematics, solving systems of linear equations in mathematics, and simplifying complex expressions in physics and engineering.

Q: Can I use the commutative property of addition to simplify the expression $(x+y+z)(yz+zx+xy)-xyz$?

A: Yes, you can use the commutative property of addition to simplify the expression $(x+y+z)(yz+zx+xy)-xyz$. We used the commutative property of addition in our previous article to rearrange the terms in the expression.

Q: How do I check my work when simplifying the expression?

A: To check your work when simplifying the expression, you need to carefully review your steps and ensure that the expression is simplified correctly. You can also use a calculator or a computer program to check your work.

Conclusion

In this article, we answered some frequently asked questions about simplifying the expression $(x+y+z)(yz+zx+xy)-xyz$. We discussed the distributive property, combining like terms, the commutative property of addition, factoring out common terms, and checking your work. We also provided some real-world applications of the expression and some tips for avoiding common mistakes.