The Variables $X, Y,$ And $Z$ Represent Polynomials Where $X=a, Y=3a-5,$ And \$Z=a^2+2$[/tex\]. What Is $X^2 Y - Z$ In Simplest Form?A. $2a^2 - 5a - 2$ B. \$3a^3 - 6a^2 -

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Understanding the Problem

In this problem, we are given three variables, X, Y, and Z, which represent polynomials. The values of these variables are defined as follows: X = a, Y = 3a - 5, and Z = a^2 + 2. We are asked to find the simplest form of the expression X^2 Y - Z.

Breaking Down the Expression

To simplify the expression X^2 Y - Z, we need to first expand the terms X^2 and Y. We can then substitute the values of X and Y into the expression and simplify.

Expanding X^2

The first step is to expand X^2. Since X = a, we can write X^2 as (a)^2, which is equal to a^2.

Expanding Y

Next, we need to expand Y. We are given that Y = 3a - 5. To expand this expression, we can multiply the term 3a by the constant -5, which gives us -15a.

Substituting Values into the Expression

Now that we have expanded X^2 and Y, we can substitute these values into the expression X^2 Y - Z. We get:

(a^2)(3a - 5) - (a^2 + 2)

Distributing a^2 to the Terms in the Parentheses

To simplify the expression, we need to distribute a^2 to the terms in the parentheses. This gives us:

3a^3 - 5a^2 - a^2 - 2

Combining Like Terms

Now that we have distributed a^2 to the terms in the parentheses, we can combine like terms. The terms -5a^2 and -a^2 are like terms, so we can combine them to get -6a^2.

Simplifying the Expression

The expression now becomes:

3a^3 - 6a^2 - 2

Conclusion

In this problem, we were given three variables, X, Y, and Z, which represent polynomials. We were asked to find the simplest form of the expression X^2 Y - Z. By expanding the terms X^2 and Y, substituting the values of X and Y into the expression, and simplifying, we found that the simplest form of the expression is 3a^3 - 6a^2 - 2.

Final Answer

The final answer is 3a^3 - 6a^2 - 2.

Discussion

This problem requires a good understanding of polynomial expressions and how to simplify them. The key to solving this problem is to expand the terms X^2 and Y, substitute the values of X and Y into the expression, and simplify. By following these steps, we can find the simplest form of the expression X^2 Y - Z.

Related Topics

  • Polynomial expressions
  • Simplifying expressions
  • Algebraic manipulation

Example Use Cases

  • Simplifying polynomial expressions in algebra
  • Finding the simplest form of an expression in calculus
  • Solving polynomial equations in mathematics

Tips and Tricks

  • Make sure to expand the terms in the expression before substituting the values of X and Y.
  • Combine like terms to simplify the expression.
  • Check your work by plugging in values for the variables to see if the expression simplifies to the expected result.

Understanding the Problem

In this article, we explored the problem of simplifying the expression X^2 Y - Z, where X = a, Y = 3a - 5, and Z = a^2 + 2. We found that the simplest form of the expression is 3a^3 - 6a^2 - 2.

Q&A

Q: What is the value of X^2?

A: The value of X^2 is a^2.

Q: What is the value of Y?

A: The value of Y is 3a - 5.

Q: How do we simplify the expression X^2 Y - Z?

A: To simplify the expression, we need to expand the terms X^2 and Y, substitute the values of X and Y into the expression, and combine like terms.

Q: What is the final answer to the problem?

A: The final answer is 3a^3 - 6a^2 - 2.

Q: What are some related topics to this problem?

A: Some related topics include polynomial expressions, simplifying expressions, and algebraic manipulation.

Q: What are some example use cases for this problem?

A: Some example use cases include simplifying polynomial expressions in algebra, finding the simplest form of an expression in calculus, and solving polynomial equations in mathematics.

Q: What are some tips and tricks for solving this problem?

A: Some tips and tricks include making sure to expand the terms in the expression before substituting the values of X and Y, combining like terms to simplify the expression, and checking your work by plugging in values for the variables to see if the expression simplifies to the expected result.

Common Mistakes

  • Failing to expand the terms in the expression before substituting the values of X and Y.
  • Not combining like terms to simplify the expression.
  • Not checking your work by plugging in values for the variables to see if the expression simplifies to the expected result.

Best Practices

  • Make sure to read the problem carefully and understand what is being asked.
  • Break down the problem into smaller steps and solve each step individually.
  • Check your work by plugging in values for the variables to see if the expression simplifies to the expected result.

Conclusion

In this article, we explored the problem of simplifying the expression X^2 Y - Z, where X = a, Y = 3a - 5, and Z = a^2 + 2. We found that the simplest form of the expression is 3a^3 - 6a^2 - 2. We also provided a Q&A section to help answer common questions and provide tips and tricks for solving this problem.

Final Answer

The final answer is 3a^3 - 6a^2 - 2.

Related Topics

  • Polynomial expressions
  • Simplifying expressions
  • Algebraic manipulation

Example Use Cases

  • Simplifying polynomial expressions in algebra
  • Finding the simplest form of an expression in calculus
  • Solving polynomial equations in mathematics

Tips and Tricks

  • Make sure to expand the terms in the expression before substituting the values of X and Y.
  • Combine like terms to simplify the expression.
  • Check your work by plugging in values for the variables to see if the expression simplifies to the expected result.