The Variables $X, Y$, And $Z$ Represent Polynomials Where $X=a, Y=3a-5$, And $ Z = A 2 + 2 Z=a^2+2 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
  • Z = a^2 + 2

We are asked to find the value of the expression X^2 Y - Z in its simplest form.

Breaking Down the Expression

To simplify the given expression, we need to follow the order of operations (PEMDAS):

  1. Evaluate the exponent (X^2)
  2. Multiply the result by Y
  3. Subtract Z from the product

Step 1: Evaluate the Exponent (X^2)

The first step is to evaluate the exponent X^2. Since X = a, we can substitute the value of X into the expression:

X^2 = (a)^2 X^2 = a^2

Step 2: Multiply the Result by Y

Next, we multiply the result by Y:

X^2 Y = a^2 (3a - 5) X^2 Y = 3a^3 - 5a^2

Step 3: Subtract Z from the Product

Finally, we subtract Z from the product:

X^2 Y - Z = (3a^3 - 5a^2) - (a^2 + 2) X^2 Y - Z = 3a^3 - 5a^2 - a^2 - 2 X^2 Y - Z = 3a^3 - 6a^2 - 2

Conclusion

In this problem, we were given three variables representing polynomials and asked to simplify the expression X^2 Y - Z. By following the order of operations and substituting the values of X, Y, and Z, we arrived at the simplified expression:

3a^3 - 6a^2 - 2

This is the correct answer, which matches option B.

Key Takeaways

  • When simplifying expressions involving polynomials, it's essential to follow the order of operations (PEMDAS).
  • Substituting the values of variables into the expression can help simplify the calculation.
  • Paying attention to the signs and exponents is crucial when simplifying polynomial expressions.

Additional Practice

To reinforce your understanding of simplifying polynomial expressions, try the following practice problems:

  • Simplify the expression (2x + 3)^2 - (x^2 + 2x + 1)
  • Evaluate the expression (x^2 - 4)(x + 2) - (x^2 + 2x + 1)
  • Simplify the expression (3x^2 - 2x + 1)(x - 1) - (x^2 + 2x + 1)

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying expressions. The acronym PEMDAS stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next (e.g., 2^3).
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, follow these steps:

  1. Evaluate any exponential expressions (e.g., 2^3).
  2. Multiply and divide any terms from left to right.
  3. Add and subtract any terms from left to right.
  4. Combine like terms (terms with the same variable and exponent).

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example:

  • 2x and 4x are like terms because they both have the variable x and the same exponent (1).
  • 3x^2 and 2x^2 are like terms because they both have the variable x and the same exponent (2).

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients (numbers in front of the variable) of the like terms. For example:

  • 2x + 4x = (2 + 4)x = 6x
  • 3x^2 - 2x^2 = (3 - 2)x^2 = x^2

Q: What is the difference between a polynomial and an algebraic expression?

A: A polynomial is an expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. An algebraic expression, on the other hand, can include any combination of variables, coefficients, and mathematical operations.

Q: How do I simplify a polynomial expression with multiple variables?

A: To simplify a polynomial expression with multiple variables, follow the same steps as before:

  1. Evaluate any exponential expressions.
  2. Multiply and divide any terms from left to right.
  3. Add and subtract any terms from left to right.
  4. Combine like terms.

When combining like terms, be sure to consider all variables and exponents.

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

A: Some common mistakes to avoid when simplifying polynomial expressions include:

  • Failing to evaluate exponential expressions first.
  • Not combining like terms correctly.
  • Forgetting to consider all variables and exponents when combining like terms.
  • Not following the order of operations (PEMDAS).

Conclusion

Simplifying polynomial expressions can be a challenging task, but by following the order of operations (PEMDAS) and combining like terms correctly, you can simplify even the most complex expressions. Remember to evaluate exponential expressions first, multiply and divide from left to right, add and subtract from left to right, and combine like terms carefully. With practice and patience, you'll become a pro at simplifying polynomial expressions in no time!