Subtract:$ \left(-7z^6 - 2\right) - \left(-3z^6\right) $Your Answer Should Be In Simplest Terms.Enter The Correct Answer.

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Introduction

Polynomial expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will focus on subtracting polynomial expressions, specifically the expression $ \left(-7z^6 - 2\right) - \left(-3z^6\right) $. We will break down the process step by step, using simple language and clear examples to ensure that you understand the concept.

What are Polynomial Expressions?

Before we dive into the subtraction process, let's quickly review what polynomial expressions are. A polynomial expression is a mathematical expression that consists of variables, coefficients, and exponents. It is a sum of terms, where each term is a product of a coefficient and a variable raised to a power.

For example, the expression 2x2+3x−42x^2 + 3x - 4 is a polynomial expression, where xx is the variable, 22, 33, and −4-4 are the coefficients, and 22, 11, and 00 are the exponents.

Subtracting Polynomial Expressions

Now that we have a basic understanding of polynomial expressions, let's move on to subtracting them. When subtracting polynomial expressions, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate any expressions inside parentheses.
  2. Exponents: Evaluate any exponents (e.g., x2x^2).
  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.

Step 1: Distribute the Negative Sign

When subtracting a polynomial expression, we need to distribute the negative sign to each term inside the parentheses. This means that we need to change the sign of each term.

For example, in the expression $ \left(-7z^6 - 2\right) - \left(-3z^6\right) $, we need to distribute the negative sign to each term inside the parentheses:

$ \left(-7z^6 - 2\right) - \left(-3z^6\right) = -7z^6 - 2 + 3z^6 $

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. Like terms are terms that have the same variable and exponent.

In our example, we have two like terms: −7z6-7z^6 and 3z63z^6. We can combine these terms by adding their coefficients:

−7z6+3z6=(−7+3)z6=−4z6-7z^6 + 3z^6 = (-7 + 3)z^6 = -4z^6

So, our expression becomes:

−4z6−2-4z^6 - 2

Step 3: Simplify the Expression

Our final step is to simplify the expression by combining any remaining like terms. In our example, we have no remaining like terms, so our expression is already simplified.

Conclusion

Subtracting polynomial expressions may seem daunting at first, but with practice and patience, you can master this skill. Remember to follow the order of operations (PEMDAS) and distribute the negative sign to each term inside the parentheses. Finally, combine like terms and simplify the expression.

By following these steps, you can simplify even the most complex polynomial expressions. So, the next time you encounter a polynomial expression, don't be afraid to give it a try!

Example Problems

Here are a few example problems to help you practice subtracting polynomial expressions:

  1. $ \left(2x^2 - 3x + 4\right) - \left(-x^2 + 2x - 3\right) $
  2. $ \left(-5z^4 + 2z^3 - 3z^2\right) - \left(3z^4 - 2z^3 + z^2\right) $
  3. $ \left(4x^3 - 2x^2 + 3x - 1\right) - \left(-x^3 + 2x^2 - 3x + 1\right) $

Answer Key

Here are the answers to the example problems:

  1. $ \left(2x^2 - 3x + 4\right) - \left(-x^2 + 2x - 3\right) = 3x^2 - 5x + 7 $
  2. $ \left(-5z^4 + 2z^3 - 3z^2\right) - \left(3z^4 - 2z^3 + z^2\right) = -8z^4 + 4z^3 - 4z^2 $
  3. $ \left(4x^3 - 2x^2 + 3x - 1\right) - \left(-x^3 + 2x^2 - 3x + 1\right) = 5x^3 - 4x^2 + 6x - 2 $

Final Thoughts

Subtracting polynomial expressions is a fundamental skill in algebra, and with practice and patience, you can master this skill. Remember to follow the order of operations (PEMDAS) and distribute the negative sign to each term inside the parentheses. Finally, combine like terms and simplify the expression.

Q: What is the order of operations (PEMDAS) and how does it apply to subtracting polynomial expressions?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. PEMDAS stands for:

  1. Parentheses: Evaluate any expressions inside parentheses.
  2. Exponents: Evaluate any exponents (e.g., x2x^2).
  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.

When subtracting polynomial expressions, we need to follow the order of operations (PEMDAS) to ensure that we perform the operations in the correct order.

Q: How do I distribute the negative sign when subtracting polynomial expressions?

A: When subtracting polynomial expressions, we need to distribute the negative sign to each term inside the parentheses. This means that we need to change the sign of each term.

For example, in the expression $ \left(-7z^6 - 2\right) - \left(-3z^6\right) $, we need to distribute the negative sign to each term inside the parentheses:

$ \left(-7z^6 - 2\right) - \left(-3z^6\right) = -7z^6 - 2 + 3z^6 $

Q: What are like terms and how do I combine them?

A: Like terms are terms that have the same variable and exponent. When we have like terms, we can combine them by adding their coefficients.

For example, in the expression −7z6+3z6-7z^6 + 3z^6, we have two like terms: −7z6-7z^6 and 3z63z^6. We can combine these terms by adding their coefficients:

−7z6+3z6=(−7+3)z6=−4z6-7z^6 + 3z^6 = (-7 + 3)z^6 = -4z^6

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, we need to combine any remaining like terms. If there are no remaining like terms, then the expression is already simplified.

For example, in the expression −4z6−2-4z^6 - 2, we have no remaining like terms, so the expression is already simplified.

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

A: Here are some common mistakes to avoid when subtracting polynomial expressions:

  • Not following the order of operations (PEMDAS)
  • Not distributing the negative sign to each term inside the parentheses
  • Not combining like terms
  • Not simplifying the expression

Q: How can I practice subtracting polynomial expressions?

A: Here are some ways to practice subtracting polynomial expressions:

  • Use online resources, such as Khan Academy or Mathway, to practice subtracting polynomial expressions.
  • Work with a tutor or teacher to practice subtracting polynomial expressions.
  • Use a calculator to check your work and ensure that you are getting the correct answer.
  • Practice subtracting polynomial expressions with different variables and exponents.

Q: What are some real-world applications of subtracting polynomial expressions?

A: Subtracting polynomial expressions has many real-world applications, including:

  • Calculating the area and perimeter of shapes
  • Determining the volume of solids
  • Modeling population growth and decline
  • Analyzing data and making predictions

Q: How can I apply subtracting polynomial expressions to my everyday life?

A: Subtracting polynomial expressions can be applied to many areas of everyday life, including:

  • Budgeting and financial planning
  • Measuring and calculating distances and times
  • Analyzing data and making predictions
  • Solving problems and making decisions

By mastering the skill of subtracting polynomial expressions, you can apply it to many areas of your life and make informed decisions.