What Is The Sum Of The Polynomials?${ (7x^3 - 4x^2) + (2x^3 - 4x^2) }$A. ${ 5x^3\$} B. ${ 9x^3\$} C. ${ 5x^3 - 8x^2\$} D. ${ 9x^3 - 8x^2\$}

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Understanding Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. They are a fundamental concept in mathematics, and understanding how to work with polynomials is crucial for solving various mathematical problems.

The Problem

We are given two polynomials:

{ (7x^3 - 4x^2) + (2x^3 - 4x^2) \}

Our task is to find the sum of these two polynomials.

Step 1: Identify Like Terms

To find the sum of the polynomials, we need to identify like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two like terms:

  • 7x37x^3 and 2x32x^3
  • −4x2-4x^2 and −4x2-4x^2

Step 2: Combine Like Terms

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

  • For the x3x^3 terms, we have 7x3+2x3=9x37x^3 + 2x^3 = 9x^3
  • For the x2x^2 terms, we have −4x2−4x2=−8x2-4x^2 - 4x^2 = -8x^2

Step 3: Write the Sum

Now that we have combined the like terms, we can write the sum of the polynomials.

{ (7x^3 - 4x^2) + (2x^3 - 4x^2) = 9x^3 - 8x^2 \}

Conclusion

The sum of the polynomials is 9x3−8x29x^3 - 8x^2. This is the correct answer.

Answer Key

The correct answer is:

{ \boxed{9x^3 - 8x^2} \}

Why is this Important?

Understanding how to work with polynomials is crucial for solving various mathematical problems. Polynomials are used in many areas of mathematics, including algebra, geometry, and calculus. By understanding how to work with polynomials, you can solve problems in these areas and gain a deeper understanding of mathematical concepts.

Real-World Applications

Polynomials have many real-world applications. For example, they are used in:

  • Physics: Polynomials are used to describe the motion of objects under the influence of forces.
  • Engineering: Polynomials are used to design and optimize systems, such as bridges and buildings.
  • Economics: Polynomials are used to model economic systems and make predictions about future economic trends.

Common Mistakes

When working with polynomials, there are several common mistakes to avoid:

  • Not identifying like terms: Failing to identify like terms can lead to incorrect answers.
  • Not combining like terms correctly: Failing to combine like terms correctly can lead to incorrect answers.
  • Not simplifying the expression: Failing to simplify the expression can lead to incorrect answers.

Tips and Tricks

Here are some tips and tricks for working with polynomials:

  • Use a systematic approach: When working with polynomials, use a systematic approach to identify like terms and combine them correctly.
  • Simplify the expression: Always simplify the expression to ensure that you have the correct answer.
  • Check your work: Always check your work to ensure that you have the correct answer.

Conclusion

Frequently Asked Questions

Q: What is the sum of the polynomials (3x^2 + 2x) + (2x^2 + 5x)?

A: To find the sum of the polynomials, we need to combine like terms. The like terms are 3x^2 and 2x^2, and 2x and 5x. Combining these terms, we get:

(3x^2 + 2x^2) + (2x + 5x) = 5x^2 + 7x

Q: What is the sum of the polynomials (x^3 - 2x^2) + (3x^3 + 4x^2)?

A: To find the sum of the polynomials, we need to combine like terms. The like terms are x^3 and 3x^3, and -2x^2 and 4x^2. Combining these terms, we get:

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

Q: What is the sum of the polynomials (2x^2 + 3x) + (x^2 - 2x)?

A: To find the sum of the polynomials, we need to combine like terms. The like terms are 2x^2 and x^2, and 3x and -2x. Combining these terms, we get:

(2x^2 + x^2) + (3x - 2x) = 3x^2 + x

Q: What is the sum of the polynomials (x^3 + 2x^2) + (3x^3 - 4x^2)?

A: To find the sum of the polynomials, we need to combine like terms. The like terms are x^3 and 3x^3, and 2x^2 and -4x^2. Combining these terms, we get:

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

Q: What is the sum of the polynomials (x^2 + 2x) + (x^2 - 3x)?

A: To find the sum of the polynomials, we need to combine like terms. The like terms are x^2 and x^2, and 2x and -3x. Combining these terms, we get:

(x^2 + x^2) + (2x - 3x) = 2x^2 - x

Q: What is the sum of the polynomials (3x^3 + 2x^2) + (x^3 - 4x^2)?

A: To find the sum of the polynomials, we need to combine like terms. The like terms are 3x^3 and x^3, and 2x^2 and -4x^2. Combining these terms, we get:

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

Conclusion

In conclusion, finding the sum of polynomials involves combining like terms. By following the steps outlined in this article, you can find the sum of polynomials and gain a deeper understanding of mathematical concepts. Remember to identify like terms, combine them correctly, and simplify the expression to ensure that you have the correct answer.

Tips and Tricks

Here are some tips and tricks for working with polynomials:

  • Use a systematic approach: When working with polynomials, use a systematic approach to identify like terms and combine them correctly.
  • Simplify the expression: Always simplify the expression to ensure that you have the correct answer.
  • Check your work: Always check your work to ensure that you have the correct answer.

Common Mistakes

When working with polynomials, there are several common mistakes to avoid:

  • Not identifying like terms: Failing to identify like terms can lead to incorrect answers.
  • Not combining like terms correctly: Failing to combine like terms correctly can lead to incorrect answers.
  • Not simplifying the expression: Failing to simplify the expression can lead to incorrect answers.

Real-World Applications

Polynomials have many real-world applications. For example, they are used in:

  • Physics: Polynomials are used to describe the motion of objects under the influence of forces.
  • Engineering: Polynomials are used to design and optimize systems, such as bridges and buildings.
  • Economics: Polynomials are used to model economic systems and make predictions about future economic trends.

Conclusion

In conclusion, understanding how to work with polynomials is crucial for solving various mathematical problems. By following the steps outlined in this article, you can find the sum of polynomials and gain a deeper understanding of mathematical concepts. Remember to identify like terms, combine them correctly, and simplify the expression to ensure that you have the correct answer.