Simplify The Expression:$\[ \left(-3 P^3 + 5 P^2 - 2 P\right) + \left(-p^3 - 8 P^2 - 15 P\right) \\]

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Introduction

In algebra, combining like terms is a fundamental concept that allows us to simplify complex expressions by adding or subtracting terms with the same variable and exponent. In this article, we will focus on simplifying the given expression by combining like terms.

The Given Expression

The given expression is:

(−3p3+5p2−2p)+(−p3−8p2−15p)\left(-3 p^3 + 5 p^2 - 2 p\right) + \left(-p^3 - 8 p^2 - 15 p\right)

Step 1: Identify Like Terms

To simplify the expression, we need to identify like terms. Like terms are terms that have the same variable and exponent. In this case, we have three types of like terms:

  • Terms with the variable p3p^3
  • Terms with the variable p2p^2
  • Terms with the variable pp

Step 2: Combine Like Terms

Now that we have identified like terms, we can combine them by adding or subtracting their coefficients.

  • For the terms with the variable p3p^3, we have −3p3-3p^3 and −p3-p^3. We can combine them by adding their coefficients: −3p3+(−p3)=−4p3-3p^3 + (-p^3) = -4p^3
  • For the terms with the variable p2p^2, we have 5p25p^2 and −8p2-8p^2. We can combine them by adding their coefficients: 5p2+(−8p2)=−3p25p^2 + (-8p^2) = -3p^2
  • For the terms with the variable pp, we have −2p-2p and −15p-15p. We can combine them by adding their coefficients: −2p+(−15p)=−17p-2p + (-15p) = -17p

Step 3: Simplify the Expression

Now that we have combined like terms, we can simplify the expression by adding the resulting terms:

−4p3−3p2−17p-4p^3 - 3p^2 - 17p

Conclusion

In this article, we have simplified the given expression by combining like terms. We identified like terms, combined them by adding or subtracting their coefficients, and simplified the expression. This process is essential in algebra, as it allows us to simplify complex expressions and make them easier to work with.

Example Use Cases

Combining like terms is a fundamental concept in algebra, and it has many practical applications. Here are a few example use cases:

  • Simplifying polynomial expressions: Combining like terms is essential in simplifying polynomial expressions, which are used to model a wide range of phenomena in mathematics, science, and engineering.
  • Solving equations: Combining like terms is also essential in solving equations, which are used to model a wide range of phenomena in mathematics, science, and engineering.
  • Graphing functions: Combining like terms is also essential in graphing functions, which are used to visualize a wide range of phenomena in mathematics, science, and engineering.

Tips and Tricks

Here are a few tips and tricks to help you simplify expressions by combining like terms:

  • Identify like terms carefully: Make sure to identify like terms carefully, as this is the first step in simplifying an expression.
  • Combine like terms systematically: Combine like terms systematically, starting with the terms with the highest exponent and working your way down.
  • Check your work: Check your work carefully to make sure that you have combined like terms correctly.

Common Mistakes

Here are a few common mistakes to avoid when simplifying expressions by combining like terms:

  • Failing to identify like terms: Failing to identify like terms is a common mistake that can lead to incorrect simplifications.
  • Combining unlike terms: Combining unlike terms is a common mistake that can lead to incorrect simplifications.
  • Not checking work: Not checking work carefully can lead to incorrect simplifications.

Final Thoughts

In conclusion, simplifying expressions by combining like terms is a fundamental concept in algebra that has many practical applications. By identifying like terms, combining them systematically, and checking your work carefully, you can simplify complex expressions and make them easier to work with. Remember to avoid common mistakes, such as failing to identify like terms, combining unlike terms, and not checking work carefully. With practice and patience, you can become proficient in simplifying expressions by combining like terms.

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Q&A: Simplifying Expressions by Combining Like Terms

Q: What are like terms in algebra?

A: Like terms are terms that have the same variable and exponent. For example, 2x22x^2 and 5x25x^2 are like terms because they both have the variable xx and the exponent 22.

Q: How do I identify like terms in an expression?

A: To identify like terms, look for terms that have the same variable and exponent. You can also use the following steps:

  1. Write down the expression.
  2. Identify the terms with the same variable and exponent.
  3. Group the like terms together.

Q: What is the difference between combining like terms and adding or subtracting terms?

A: Combining like terms is a process of adding or subtracting terms with the same variable and exponent. Adding or subtracting terms is a process of combining terms with different variables or exponents.

Q: Can I combine unlike terms?

A: No, you cannot combine unlike terms. Unlike terms are terms that have different variables or exponents. Combining unlike terms can lead to incorrect simplifications.

Q: How do I combine like terms?

A: To combine like terms, follow these steps:

  1. Identify the like terms.
  2. Add or subtract the coefficients of the like terms.
  3. Write the resulting term.

Q: What is the coefficient of a term?

A: The coefficient of a term is the number that is multiplied by the variable. For example, in the term 3x3x, the coefficient is 33.

Q: Can I simplify an expression by combining like terms if it has multiple variables?

A: Yes, you can simplify an expression by combining like terms if it has multiple variables. However, you need to identify the like terms carefully and combine them systematically.

Q: How do I check my work when simplifying an expression by combining like terms?

A: To check your work, follow these steps:

  1. Write down the original expression.
  2. Simplify the expression by combining like terms.
  3. Compare the simplified expression with the original expression.

Q: What are some common mistakes to avoid when simplifying expressions by combining like terms?

A: Some common mistakes to avoid when simplifying expressions by combining like terms include:

  • Failing to identify like terms
  • Combining unlike terms
  • Not checking work carefully

Q: Can I use a calculator to simplify expressions by combining like terms?

A: Yes, you can use a calculator to simplify expressions by combining like terms. However, it's always a good idea to check your work manually to ensure that you have combined like terms correctly.

Q: How do I apply the concept of combining like terms to real-world problems?

A: The concept of combining like terms can be applied to real-world problems in a variety of ways, such as:

  • Simplifying polynomial expressions to model population growth or chemical reactions
  • Solving equations to model financial problems or physical systems
  • Graphing functions to visualize data or model real-world phenomena

Q: Can I use the concept of combining like terms to simplify expressions with negative coefficients?

A: Yes, you can use the concept of combining like terms to simplify expressions with negative coefficients. When combining like terms with negative coefficients, be sure to add or subtract the coefficients carefully.

Q: How do I simplify expressions with variables raised to different powers?

A: To simplify expressions with variables raised to different powers, follow these steps:

  1. Identify the like terms.
  2. Combine the like terms by adding or subtracting their coefficients.
  3. Write the resulting term.

Q: Can I use the concept of combining like terms to simplify expressions with fractions?

A: Yes, you can use the concept of combining like terms to simplify expressions with fractions. When combining like terms with fractions, be sure to add or subtract the numerators and denominators carefully.

Q: How do I apply the concept of combining like terms to simplify expressions with multiple variables?

A: To apply the concept of combining like terms to simplify expressions with multiple variables, follow these steps:

  1. Identify the like terms.
  2. Combine the like terms by adding or subtracting their coefficients.
  3. Write the resulting term.

Q: Can I use the concept of combining like terms to simplify expressions with exponents?

A: Yes, you can use the concept of combining like terms to simplify expressions with exponents. When combining like terms with exponents, be sure to add or subtract the exponents carefully.

Q: How do I check my work when simplifying expressions with multiple variables?

A: To check your work when simplifying expressions with multiple variables, follow these steps:

  1. Write down the original expression.
  2. Simplify the expression by combining like terms.
  3. Compare the simplified expression with the original expression.

Q: What are some real-world applications of the concept of combining like terms?

A: Some real-world applications of the concept of combining like terms include:

  • Simplifying polynomial expressions to model population growth or chemical reactions
  • Solving equations to model financial problems or physical systems
  • Graphing functions to visualize data or model real-world phenomena

Q: Can I use the concept of combining like terms to simplify expressions with radicals?

A: Yes, you can use the concept of combining like terms to simplify expressions with radicals. When combining like terms with radicals, be sure to add or subtract the coefficients carefully.

Q: How do I apply the concept of combining like terms to simplify expressions with absolute values?

A: To apply the concept of combining like terms to simplify expressions with absolute values, follow these steps:

  1. Identify the like terms.
  2. Combine the like terms by adding or subtracting their coefficients.
  3. Write the resulting term.

Q: Can I use the concept of combining like terms to simplify expressions with complex numbers?

A: Yes, you can use the concept of combining like terms to simplify expressions with complex numbers. When combining like terms with complex numbers, be sure to add or subtract the real and imaginary parts carefully.

Q: How do I check my work when simplifying expressions with complex numbers?

A: To check your work when simplifying expressions with complex numbers, follow these steps:

  1. Write down the original expression.
  2. Simplify the expression by combining like terms.
  3. Compare the simplified expression with the original expression.

Q: What are some common mistakes to avoid when simplifying expressions with complex numbers?

A: Some common mistakes to avoid when simplifying expressions with complex numbers include:

  • Failing to identify like terms
  • Combining unlike terms
  • Not checking work carefully