Evaluating 2x³y - 3x²y²z + 2xy³z For X=-2, Y=1, Z=0

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In this article, we will delve into the process of evaluating a mathematical expression given specific values for its variables. Specifically, we will focus on the expression 2x³y - 3x²y²z + 2xy³z, where x = -2, y = 1, and z = 0. This type of problem is a fundamental concept in algebra and is crucial for understanding more complex mathematical concepts. By carefully substituting the given values and performing the necessary arithmetic operations, we can arrive at the solution. This exercise not only reinforces our understanding of algebraic manipulation but also highlights the importance of order of operations and attention to detail in mathematical calculations.

Understanding the Problem

Before we begin the evaluation, let's break down the problem and understand the key components. The expression 2x³y - 3x²y²z + 2xy³z is a polynomial expression consisting of three terms. Each term involves variables x, y, and z raised to certain powers and multiplied by coefficients. Our task is to substitute the given values, x = -2, y = 1, and z = 0, into the expression and simplify it to obtain a numerical value. Understanding the structure of the expression and the order of operations (PEMDAS/BODMAS) is crucial for accurate evaluation. The presence of exponents and multiple variables requires careful attention to detail to avoid errors. By systematically substituting the values and performing the calculations, we can successfully evaluate the expression.

Breaking Down the Expression

The expression 2x³y - 3x²y²z + 2xy³z can be broken down into three distinct terms, each requiring separate evaluation before being combined. Let's analyze each term individually:

  1. 2x³y: This term involves the variable x raised to the power of 3, multiplied by the variable y, and then multiplied by the coefficient 2. The exponentiation operation (x³) needs to be performed before multiplication. Understanding the impact of a negative value for x on the result of x³ is crucial.
  2. -3x²y²z: This term involves x squared, y squared, and z, all multiplied together and then multiplied by the coefficient -3. Here, both x and y are raised to the power of 2, and their product is further multiplied by z. The presence of z in this term is significant, as its value can greatly impact the overall result.
  3. 2xy³z: This term includes x, y cubed, and z, multiplied together and then by the coefficient 2. Similar to the previous term, y is raised to a power before multiplication. The presence of z also plays a critical role in determining the value of this term.

By dissecting the expression in this manner, we can approach the evaluation systematically, ensuring that each component is calculated correctly before combining them.

Importance of Order of Operations

The order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), is paramount in evaluating mathematical expressions accurately. It dictates the sequence in which operations must be performed to arrive at the correct result. In the expression 2x³y - 3x²y²z + 2xy³z, the order of operations is crucial. First, we must handle the exponents (x³, x², y³). Next, we perform the multiplications within each term. Finally, we carry out the addition and subtraction to combine the terms. Neglecting the order of operations can lead to significant errors in the final result. For instance, multiplying before exponentiating would yield an incorrect value. Therefore, strictly adhering to PEMDAS/BODMAS is essential for accurate evaluation. Understanding and applying the order of operations correctly demonstrates a solid foundation in algebraic principles.

Step-by-Step Evaluation

Now, let's proceed with the step-by-step evaluation of the expression 2x³y - 3x²y²z + 2xy³z given x = -2, y = 1, and z = 0. We will follow the order of operations (PEMDAS/BODMAS) to ensure accuracy.

Step 1: Substitute the Values

The first step is to substitute the given values of x, y, and z into the expression. This replaces the variables with their corresponding numerical values, allowing us to proceed with the arithmetic operations. Substituting x = -2, y = 1, and z = 0 into 2x³y - 3x²y²z + 2xy³z yields:

2(-2)³(1) - 3(-2)²(1)²(0) + 2(-2)(1)³(0)

This substitution is a critical step, transforming the algebraic expression into a numerical one that can be simplified. Accuracy in substitution is paramount, as any error at this stage will propagate through the rest of the calculation. By carefully replacing each variable with its assigned value, we set the stage for the subsequent steps in the evaluation process.

Step 2: Evaluate the Exponents

Following the order of operations, we address the exponents in the expression. We need to calculate (-2)³, (-2)², and (1)³. Let's break down each calculation:

  • (-2)³ = (-2) * (-2) * (-2) = -8
  • (-2)² = (-2) * (-2) = 4
  • (1)³ = 1 * 1 * 1 = 1

Substituting these values back into the expression, we get:

2(-8)(1) - 3(4)(1)²(0) + 2(-2)(1)(0)

Evaluating exponents is a crucial step, as they define the magnitude of the terms. Understanding how negative numbers behave when raised to different powers is essential for correct calculations. By accurately evaluating the exponents, we simplify the expression further, bringing us closer to the final solution.

Step 3: Perform the Multiplications

Next, we perform the multiplications within each term of the expression. We have three terms to consider:

  1. 2(-8)(1) = -16
  2. -3(4)(1)²(0) = -3(4)(1)(0) = 0
  3. 2(-2)(1)(0) = -4(1)(0) = 0

Substituting these results back into the expression, we now have:

-16 - 0 + 0

Multiplication is a fundamental arithmetic operation, and it is crucial to perform it accurately in each term. The presence of zero as a factor in the second and third terms significantly simplifies the expression, as any number multiplied by zero is zero. By carefully performing the multiplications, we reduce the expression to a simple addition and subtraction problem.

Step 4: Perform the Addition and Subtraction

Finally, we perform the addition and subtraction operations to obtain the final result. The expression now simplifies to:

-16 - 0 + 0 = -16

Therefore, the value of the expression 2x³y - 3x²y²z + 2xy³z when x = -2, y = 1, and z = 0 is -16.

Addition and subtraction are the final steps in evaluating the expression. After performing all the necessary multiplications and exponentiations, these operations combine the terms to produce the final numerical value. In this case, the subtraction of zero does not change the value, and the final result is simply -16. This concludes the evaluation process, providing the solution to the problem.

Final Result

After carefully following the order of operations and performing each step meticulously, we have arrived at the final result. The value of the expression 2x³y - 3x²y²z + 2xy³z when x = -2, y = 1, and z = 0 is -16. This result is obtained by substituting the given values, evaluating the exponents, performing the multiplications, and finally, carrying out the addition and subtraction. The step-by-step approach ensures accuracy and demonstrates a thorough understanding of algebraic evaluation.

Common Mistakes to Avoid

Evaluating algebraic expressions can be challenging, and it's easy to make mistakes if you're not careful. Here are some common pitfalls to avoid when evaluating expressions like 2x³y - 3x²y²z + 2xy³z:

  • Incorrect Order of Operations: As discussed earlier, following the order of operations (PEMDAS/BODMAS) is crucial. A common mistake is to multiply before evaluating exponents or to add/subtract before multiplying. Always prioritize exponents first, followed by multiplication and division, and then addition and subtraction.
  • Sign Errors: Dealing with negative numbers can be tricky. Pay close attention to the signs when substituting values and performing calculations. For instance, (-2)³ is -8, while (-2)² is 4. Incorrectly handling signs can lead to significant errors in the final result.
  • Substitution Errors: Ensure that you substitute the values correctly. Double-check that you've replaced each variable with its corresponding value. A simple mistake in substitution can throw off the entire calculation.
  • Forgetting the Zero Property: Remember that any term multiplied by zero equals zero. In the expression 2x³y - 3x²y²z + 2xy³z, the terms containing z become zero when z = 0. Neglecting this property can lead to unnecessary calculations and potential errors.
  • Arithmetic Errors: Simple arithmetic errors, such as miscalculating a multiplication or addition, can occur. Take your time and double-check your calculations to minimize these errors.

By being aware of these common mistakes and taking precautions to avoid them, you can improve your accuracy in evaluating algebraic expressions.

Conclusion

In conclusion, we have successfully evaluated the expression 2x³y - 3x²y²z + 2xy³z given x = -2, y = 1, and z = 0. By following a systematic approach, which included substituting the values, evaluating exponents, performing multiplications, and carrying out addition and subtraction, we arrived at the final result of -16. This exercise highlights the importance of understanding algebraic principles, adhering to the order of operations, and paying attention to detail. Evaluating algebraic expressions is a fundamental skill in mathematics, and mastering it is essential for success in more advanced topics. By practicing and being mindful of common mistakes, you can confidently tackle similar problems and strengthen your mathematical abilities. The process of breaking down the expression into smaller parts, addressing each component methodically, and then combining the results is a valuable strategy applicable to various mathematical problems. This methodical approach not only ensures accuracy but also enhances understanding and problem-solving skills. Therefore, the ability to evaluate expressions like this is a cornerstone of mathematical proficiency.