Subtract These Polynomials:$\left(6x^2 - X + 8\right) - \left(x^2 + 2\right) =$A. $5x^2 - X + 6$B. $5x^2 - X + 10$C. $7x^2 - X + 10$D. $7x^2 - X + 6$

by ADMIN 150 views

=====================================================

Introduction

Polynomials are a fundamental concept in algebra, and subtracting them is an essential operation in mathematics. In this article, we will explore the process of subtracting polynomials, focusing on the given problem: (6x2βˆ’x+8)βˆ’(x2+2)=\left(6x^2 - x + 8\right) - \left(x^2 + 2\right) =A. 5x2βˆ’x+65x^2 - x + 6B. 5x2βˆ’x+105x^2 - x + 10C. 7x2βˆ’x+107x^2 - x + 10D. 7x2βˆ’x+67x^2 - x + 6. We will break down the solution step by step, providing a clear understanding of the process.

What are Polynomials?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be written in the form anxn+anβˆ’1xnβˆ’1+…+a1x+a0a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0, where anβ‰ 0a_n \neq 0. The degree of a polynomial is the highest power of the variable.

Subtracting Polynomials

Subtracting polynomials involves combining like terms, which are terms with the same variable and exponent. To subtract polynomials, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  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-by-Step Solution

Let's apply the steps above to the given problem:

(6x2βˆ’x+8)βˆ’(x2+2)\left(6x^2 - x + 8\right) - \left(x^2 + 2\right)

Step 1: Distribute the Negative Sign

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

βˆ’(x2+2)=βˆ’x2βˆ’2- \left(x^2 + 2\right) = -x^2 - 2

Step 2: Combine Like Terms

Now, we can combine like terms by adding or subtracting the coefficients of the same variable and exponent:

6x2βˆ’x+8βˆ’x2βˆ’26x^2 - x + 8 - x^2 - 2

Step 3: Simplify the Expression

Combine like terms:

6x2βˆ’x2βˆ’x+8βˆ’26x^2 - x^2 - x + 8 - 2

=5x2βˆ’x+6= 5x^2 - x + 6

Conclusion

In conclusion, subtracting polynomials involves combining like terms and following the order of operations. By applying the steps above, we can simplify the given expression and arrive at the correct solution: 5x2βˆ’x+65x^2 - x + 6. This demonstrates the importance of understanding polynomial operations and how they can be applied to solve mathematical problems.

Frequently Asked Questions

Q: What is the difference between adding and subtracting polynomials?

A: Adding polynomials involves combining like terms by adding the coefficients of the same variable and exponent. Subtracting polynomials involves combining like terms by subtracting the coefficients of the same variable and exponent.

Q: How do I know which terms to combine when subtracting polynomials?

A: When subtracting polynomials, combine like terms by adding or subtracting the coefficients of the same variable and exponent.

Q: Can I simplify polynomials using other methods?

A: Yes, polynomials can be simplified using other methods, such as factoring or using the distributive property.

Final Answer

The final answer is 5x2βˆ’x+6\boxed{5x^2 - x + 6}.

Additional Resources

For more information on polynomials and their operations, check out the following resources:

  • Khan Academy: Polynomials
  • Mathway: Subtracting Polynomials
  • Wolfram Alpha: Polynomials

By following the steps outlined in this article, you should now have a clear understanding of how to subtract polynomials. Remember to always combine like terms and follow the order of operations to simplify expressions.

=====================================================

Introduction

Subtracting polynomials is a fundamental operation in algebra, and it's essential to understand the process to solve mathematical problems. In this article, we will address some of the most frequently asked questions about subtracting polynomials, providing clear and concise answers to help you better understand the concept.

Q&A

Q: What is the difference between adding and subtracting polynomials?

A: Adding polynomials involves combining like terms by adding the coefficients of the same variable and exponent. Subtracting polynomials involves combining like terms by subtracting the coefficients of the same variable and exponent.

Q: How do I know which terms to combine when subtracting polynomials?

A: When subtracting polynomials, combine like terms by adding or subtracting the coefficients of the same variable and exponent. For example, in the expression 6x2βˆ’x+8βˆ’x2βˆ’26x^2 - x + 8 - x^2 - 2, combine the like terms 6x26x^2 and βˆ’x2-x^2 to get 5x25x^2.

Q: Can I simplify polynomials using other methods?

A: Yes, polynomials can be simplified using other methods, such as factoring or using the distributive property. Factoring involves expressing a polynomial as a product of simpler polynomials, while the distributive property involves multiplying a polynomial by a monomial.

Q: How do I handle negative coefficients when subtracting polynomials?

A: When subtracting polynomials, negative coefficients are handled by changing the sign of the term. For example, in the expression βˆ’x2βˆ’2-x^2 - 2, the negative sign is distributed to the term βˆ’x2-x^2, resulting in βˆ’x2+2-x^2 + 2.

Q: Can I subtract polynomials with different degrees?

A: Yes, polynomials with different degrees can be subtracted. However, the resulting polynomial will have the same degree as the polynomial with the higher degree.

Q: How do I simplify polynomials with multiple variables?

A: When simplifying polynomials with multiple variables, combine like terms by adding or subtracting the coefficients of the same variable and exponent. For example, in the expression 2x2yβˆ’3x2y+4xy2x^2y - 3x^2y + 4xy, combine the like terms 2x2y2x^2y and βˆ’3x2y-3x^2y to get βˆ’x2y-x^2y, and then combine the like terms βˆ’x2y-x^2y and 4xy4xy to get 3xy3xy.

Q: Can I use technology to simplify polynomials?

A: Yes, technology can be used to simplify polynomials. Many calculators and computer algebra systems, such as Wolfram Alpha, can simplify polynomials and perform other mathematical operations.

Conclusion

In conclusion, subtracting polynomials is a fundamental operation in algebra, and it's essential to understand the process to solve mathematical problems. By following the steps outlined in this article and addressing the frequently asked questions, you should now have a clear understanding of how to subtract polynomials.

Additional Resources

For more information on polynomials and their operations, check out the following resources:

  • Khan Academy: Polynomials
  • Mathway: Subtracting Polynomials
  • Wolfram Alpha: Polynomials

By following the steps outlined in this article and using the additional resources provided, you should be able to simplify polynomials and perform other mathematical operations with confidence.

Final Answer

The final answer is 5x2βˆ’x+6\boxed{5x^2 - x + 6}.

Final Tips

  • Always combine like terms when subtracting polynomials.
  • Follow the order of operations (PEMDAS) when simplifying polynomials.
  • Use technology, such as calculators and computer algebra systems, to simplify polynomials and perform other mathematical operations.

By following these tips and using the resources provided, you should be able to simplify polynomials and perform other mathematical operations with confidence.