Find The Sum Or Difference. Write Your Answer In Standard Form.$\left(2c^2 + 6c + 4\right) + \left(5c^2 - 7\right$\]

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Introduction

In algebra, finding the sum or difference of two or more polynomials is a fundamental operation. It involves combining like terms to simplify the expression. In this article, we will focus on finding the sum of two given polynomials, (2c2+6c+4)\left(2c^2 + 6c + 4\right) and (5c2−7)\left(5c^2 - 7\right). We will use the concept of like terms and the rules of algebra to simplify the expression and write the answer in standard form.

Understanding Like Terms

Like terms are terms that have the same variable raised to the same power. In the given polynomials, the like terms are the terms with the variable cc raised to the power of 2, the terms with the variable cc raised to the power of 1, and the constant terms.

Combining Like Terms

To find the sum of the two polynomials, we need to combine the like terms. We will start by combining the terms with the variable cc raised to the power of 2.

Combining the Terms with c2c^2

The terms with c2c^2 are 2c22c^2 and 5c25c^2. To combine these terms, we add their coefficients, which are 2 and 5, respectively.

# Importing necessary modules
import sympy as sp

# Defining the variable
c = sp.symbols('c')

# Defining the terms
term1 = 2*c**2
term2 = 5*c**2

# Combining the terms
combined_term = term1 + term2

print(combined_term)

The output of the above code is 7c27c^2. Therefore, the combined term is 7c27c^2.

Combining the Terms with cc

The terms with cc are 6c6c and −7c-7c. To combine these terms, we add their coefficients, which are 6 and -7, respectively.

# Importing necessary modules
import sympy as sp

# Defining the variable
c = sp.symbols('c')

# Defining the terms
term1 = 6*c
term2 = -7*c

# Combining the terms
combined_term = term1 + term2

print(combined_term)

The output of the above code is −c-c. Therefore, the combined term is −c-c.

Combining the Constant Terms

The constant terms are 4 and -7. To combine these terms, we add their coefficients, which are 4 and -7, respectively.

# Importing necessary modules
import sympy as sp

# Defining the variable
c = sp.symbols('c')

# Defining the terms
term1 = 4
term2 = -7

# Combining the terms
combined_term = term1 + term2

print(combined_term)

The output of the above code is −3-3. Therefore, the combined term is −3-3.

Writing the Answer in Standard Form

Now that we have combined the like terms, we can write the answer in standard form. The standard form of a polynomial is the form in which the terms are arranged in descending order of the powers of the variable.

The final answer is 7c2−c−3\boxed{7c^2 - c - 3}.

Conclusion

In this article, we learned how to find the sum of two polynomials by combining like terms. We used the concept of like terms and the rules of algebra to simplify the expression and write the answer in standard form. We also used Python code to demonstrate the process of combining like terms. The final answer is 7c2−c−3\boxed{7c^2 - c - 3}.

Frequently Asked Questions

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power.

Q: How do I combine like terms?

A: To combine like terms, you add their coefficients.

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is the form in which the terms are arranged in descending order of the powers of the variable.

Q: How do I write the answer in standard form?

A: To write the answer in standard form, you arrange the terms in descending order of the powers of the variable.

Further Reading

If you want to learn more about finding the sum or difference of polynomials, you can check out the following resources:

  • Algebra.com: This website has a comprehensive guide to algebra, including a section on polynomials.
  • Mathway.com: This website has a math problem solver that can help you solve algebra problems, including finding the sum or difference of polynomials.
  • KhanAcademy.org: This website has a video series on algebra, including a section on polynomials.

References

  • Algebra.com: This website has a comprehensive guide to algebra, including a section on polynomials.
  • Mathway.com: This website has a math problem solver that can help you solve algebra problems, including finding the sum or difference of polynomials.
  • KhanAcademy.org: This website has a video series on algebra, including a section on polynomials.

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Introduction

In our previous article, we learned how to find the sum of two polynomials by combining like terms. In this article, we will answer some frequently asked questions about finding the sum or difference of polynomials.

Q&A

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power.

Q: How do I combine like terms?

A: To combine like terms, you add their coefficients.

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is the form in which the terms are arranged in descending order of the powers of the variable.

Q: How do I write the answer in standard form?

A: To write the answer in standard form, you arrange the terms in descending order of the powers of the variable.

Q: Can I use a calculator to find the sum or difference of polynomials?

A: Yes, you can use a calculator to find the sum or difference of polynomials. However, it's always a good idea to check your work by hand to make sure you understand the process.

Q: How do I handle negative coefficients when combining like terms?

A: When combining like terms with negative coefficients, you simply add the coefficients as you would with positive coefficients. For example, if you have -3x and 2x, you would combine them to get -x.

Q: Can I use the distributive property to find the sum or difference of polynomials?

A: Yes, you can use the distributive property to find the sum or difference of polynomials. The distributive property states that a(b + c) = ab + ac.

Q: How do I handle polynomials with multiple variables?

A: When working with polynomials with multiple variables, you need to combine like terms for each variable separately. For example, if you have 2x^2y and 3x^2y, you would combine them to get 5x^2y.

Q: Can I use a graphing calculator to find the sum or difference of polynomials?

A: Yes, you can use a graphing calculator to find the sum or difference of polynomials. However, it's always a good idea to check your work by hand to make sure you understand the process.

Conclusion

In this article, we answered some frequently asked questions about finding the sum or difference of polynomials. We covered topics such as like terms, combining like terms, standard form, and handling negative coefficients. We also discussed using calculators and graphing calculators to find the sum or difference of polynomials.

Further Reading

If you want to learn more about finding the sum or difference of polynomials, you can check out the following resources:

  • Algebra.com: This website has a comprehensive guide to algebra, including a section on polynomials.
  • Mathway.com: This website has a math problem solver that can help you solve algebra problems, including finding the sum or difference of polynomials.
  • KhanAcademy.org: This website has a video series on algebra, including a section on polynomials.

References

  • Algebra.com: This website has a comprehensive guide to algebra, including a section on polynomials.
  • Mathway.com: This website has a math problem solver that can help you solve algebra problems, including finding the sum or difference of polynomials.
  • KhanAcademy.org: This website has a video series on algebra, including a section on polynomials.

Additional Resources

  • Polynomial Addition and Subtraction: This website has a detailed explanation of polynomial addition and subtraction.
  • Polynomial Operations: This website has a comprehensive guide to polynomial operations, including addition and subtraction.
  • Algebra Help: This website has a math problem solver that can help you solve algebra problems, including finding the sum or difference of polynomials.