What Is The Difference Of The Polynomials?$\left(5x^3 + 4x^2\right) - \left(6x^2 - 2x - 9\right$\]A. $-x^3 + 6x^2 + 9$ B. $-x^3 + 2x^2 - 9$ C. $5x^3 - 2x^2 - 2x - 9$ D. $5x^3 - 2x^2 + 2x + 9$

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Understanding Polynomials and Their Operations

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 their operations is crucial for solving various mathematical problems. In this article, we will explore the difference of polynomials, which is a fundamental operation in algebra.

What is the Difference of Polynomials?

The difference of polynomials is a mathematical operation that involves subtracting one polynomial from another. It is denoted by the symbol - and is used to find the result of subtracting one polynomial from another. The difference of polynomials is an essential concept in algebra, and it has numerous applications in various fields, including mathematics, physics, and engineering.

Step-by-Step Solution

To find the difference of the given polynomials, we need to follow a step-by-step approach. The given polynomials are:

(5x3+4x2)−(6x2−2x−9)\left(5x^3 + 4x^2\right) - \left(6x^2 - 2x - 9\right)

Step 1: Distribute the Negative Sign

The first step is to distribute the negative sign to each term in the second polynomial. This will change the sign of each term in the second polynomial.

(5x3+4x2)−(6x2−2x−9)=5x3+4x2−6x2+2x+9\left(5x^3 + 4x^2\right) - \left(6x^2 - 2x - 9\right) = 5x^3 + 4x^2 - 6x^2 + 2x + 9

Step 2: Combine Like Terms

The next step is to combine like terms. Like terms are terms that have the same variable and exponent. In this case, we have two like terms: 4x24x^2 and −6x2-6x^2. We can combine these terms by adding their coefficients.

5x3+4x2−6x2+2x+9=5x3−2x2+2x+95x^3 + 4x^2 - 6x^2 + 2x + 9 = 5x^3 - 2x^2 + 2x + 9

Step 3: Simplify the Expression

The final step is to simplify the expression by combining any remaining like terms. In this case, there are no remaining like terms, so the expression is already simplified.

Conclusion

In conclusion, the difference of the given polynomials is 5x3−2x2+2x+95x^3 - 2x^2 + 2x + 9. This result can be verified by plugging in values for xx and evaluating the expression.

Answer

The correct answer is:

D. 5x3−2x2+2x+95x^3 - 2x^2 + 2x + 9

Why is this the Correct Answer?

This is the correct answer because it is the result of subtracting the second polynomial from the first polynomial. The difference of polynomials is a fundamental operation in algebra, and it is essential to understand how to perform this operation correctly.

Real-World Applications

The difference of polynomials has numerous real-world applications in various fields, including mathematics, physics, and engineering. For example, in physics, the difference of polynomials can be used to model the motion of objects and to solve problems involving forces and energies. In engineering, the difference of polynomials can be used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

In conclusion, the difference of polynomials is a fundamental operation in algebra that involves subtracting one polynomial from another. It is essential to understand how to perform this operation correctly, and it has numerous real-world applications in various fields. By following the step-by-step approach outlined in this article, you can find the difference of any two polynomials.

Frequently Asked Questions

Q: What is the difference of polynomials?

A: The difference of polynomials is a mathematical operation that involves subtracting one polynomial from another.

Q: How do I find the difference of two polynomials?

A: To find the difference of two polynomials, you need to follow a step-by-step approach. First, distribute the negative sign to each term in the second polynomial. Then, combine like terms. Finally, simplify the expression.

Q: What are the real-world applications of the difference of polynomials?

A: The difference of polynomials has numerous real-world applications in various fields, including mathematics, physics, and engineering.

Q: Why is it essential to understand the difference of polynomials?

A: It is essential to understand the difference of polynomials because it is a fundamental operation in algebra that has numerous real-world applications.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

Additional Resources

  • [1] Khan Academy: Polynomials
  • [2] MIT OpenCourseWare: Algebra
  • [3] Wolfram Alpha: Polynomials
    Frequently Asked Questions About Polynomials =====================================================

Q&A Article

Polynomials are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. In this article, we will answer some frequently asked questions about polynomials.

Q: What is a polynomial?

A: A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What are the different types of polynomials?

A: There are several types of polynomials, including:

  • Monomials: A monomial is a polynomial with only one term. For example, 3x is a monomial.
  • Binomials: A binomial is a polynomial with two terms. For example, 3x + 2y is a binomial.
  • Trinomials: A trinomial is a polynomial with three terms. For example, 3x + 2y - 4z is a trinomial.
  • Polynomials of degree n: A polynomial of degree n is a polynomial with n terms.

Q: How do I add polynomials?

A: To add polynomials, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, to add 3x + 2y and 4x + 5y, you would combine the like terms to get 7x + 7y.

Q: How do I subtract polynomials?

A: To subtract polynomials, you need to distribute the negative sign to each term in the second polynomial. Then, combine like terms. For example, to subtract 4x + 5y from 3x + 2y, you would distribute the negative sign to get 3x + 2y - 4x - 5y. Then, combine the like terms to get -x - 3y.

Q: How do I multiply polynomials?

A: To multiply polynomials, you need to use the distributive property. The distributive property states that a(b + c) = ab + ac. For example, to multiply 3x + 2y by 4x + 5y, you would use the distributive property to get 12x^2 + 15xy + 8xy + 10y^2. Then, combine the like terms to get 12x^2 + 23xy + 10y^2.

Q: What are the real-world applications of polynomials?

A: Polynomials have numerous real-world applications in various fields, including mathematics, physics, and engineering. For example, in physics, polynomials can be used to model the motion of objects and to solve problems involving forces and energies. In engineering, polynomials can be used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: Why is it essential to understand polynomials?

A: It is essential to understand polynomials because they are a fundamental concept in mathematics and have numerous real-world applications. Understanding polynomials can help you to solve various mathematical problems and to apply mathematical concepts to real-world situations.

Q: What are some common mistakes to avoid when working with polynomials?

A: Some common mistakes to avoid when working with polynomials include:

  • Not combining like terms: Failing to combine like terms can lead to incorrect results.
  • Not using the distributive property: Failing to use the distributive property can lead to incorrect results.
  • Not simplifying expressions: Failing to simplify expressions can lead to incorrect results.

Conclusion

In conclusion, polynomials are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. By following the steps outlined in this article, you can answer frequently asked questions about polynomials and apply mathematical concepts to real-world situations.

Frequently Asked Questions

Q: What is a polynomial?

A: A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What are the different types of polynomials?

A: There are several types of polynomials, including monomials, binomials, trinomials, and polynomials of degree n.

Q: How do I add polynomials?

A: To add polynomials, you need to combine like terms.

Q: How do I subtract polynomials?

A: To subtract polynomials, you need to distribute the negative sign to each term in the second polynomial. Then, combine like terms.

Q: How do I multiply polynomials?

A: To multiply polynomials, you need to use the distributive property.

Q: What are the real-world applications of polynomials?

A: Polynomials have numerous real-world applications in various fields, including mathematics, physics, and engineering.

Q: Why is it essential to understand polynomials?

A: It is essential to understand polynomials because they are a fundamental concept in mathematics and have numerous real-world applications.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

Additional Resources

  • [1] Khan Academy: Polynomials
  • [2] MIT OpenCourseWare: Algebra
  • [3] Wolfram Alpha: Polynomials