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

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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, using the given example: (6x2−x+8)−(x2+2)\left(6x^2 - x + 8\right) - \left(x^2 + 2\right). We will break down the steps involved in subtracting polynomials and provide a clear explanation 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 of axn+bxn−1+…+cx+dax^n + bx^{n-1} + \ldots + cx + d, where aa, bb, …\ldots, cc, and dd are constants, and xx is the variable.

Subtracting Polynomials

Subtracting polynomials involves combining like terms, which means combining 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 example: (6x2−x+8)−(x2+2)\left(6x^2 - x + 8\right) - \left(x^2 + 2\right).

Step 1: Evaluate Expressions Inside Parentheses

We have two expressions inside parentheses: (6x2−x+8)\left(6x^2 - x + 8\right) and (x2+2)\left(x^2 + 2\right). We will leave these expressions as they are for now.

Step 2: Combine Like Terms

Now, we need to combine like terms. We have two terms with the same variable and exponent: 6x26x^2 and −x2-x^2. We can combine these terms by adding their coefficients: 6x2−x2=5x26x^2 - x^2 = 5x^2.

Step 3: Combine Remaining Terms

We have two remaining terms: −x-x and +2+2. We can combine these terms by adding their coefficients: −x+2=−x+2-x + 2 = -x + 2.

Step 4: Simplify the Expression

Now, we can simplify the expression by combining the like terms: 5x2−x+25x^2 - x + 2.

Conclusion

In conclusion, subtracting polynomials involves combining like terms and following the order of operations. By applying the steps above, we can simplify the expression (6x2−x+8)−(x2+2)\left(6x^2 - x + 8\right) - \left(x^2 + 2\right) to 5x2−x+25x^2 - x + 2.

Answer

The correct answer is A. ${5x^2 - x + 10}$ is incorrect, the correct answer is A. ${5x^2 - x + 2}$

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

A: The main difference between subtracting polynomials and adding polynomials is the operation being performed. When subtracting polynomials, we are combining like terms and performing a subtraction operation, whereas when adding polynomials, we are combining like terms and performing an addition operation.

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

A: When subtracting polynomials, you need to combine like terms, which means combining terms with the same variable and exponent. To do this, you need to identify the terms with the same variable and exponent and then combine their coefficients.

Q: What is the order of operations when subtracting polynomials?

A: The order of operations when subtracting polynomials is the same as when adding polynomials:

  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.

Q: Can I subtract a polynomial from a non-polynomial expression?

A: No, you cannot subtract a polynomial from a non-polynomial expression. When subtracting polynomials, both expressions must be polynomials.

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

A: When subtracting polynomials, negative coefficients are handled in the same way as positive coefficients. You simply combine the like terms and perform the subtraction operation.

Q: Can I use a calculator to subtract polynomials?

A: Yes, you can use a calculator to subtract polynomials. However, it's always a good idea to double-check your work by hand to ensure that the calculation is correct.

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

A: Some common mistakes to avoid when subtracting polynomials include:

  • Forgetting to combine like terms
  • Not following the order of operations
  • Subtracting a polynomial from a non-polynomial expression
  • Not handling negative coefficients correctly

Q: How do I check my work when subtracting polynomials?

A: To check your work when subtracting polynomials, you can:

  • Plug in a value for the variable and evaluate the expression
  • Use a calculator to check the calculation
  • Simplify the expression and check that it matches the expected result

Q: Can I use algebraic properties to simplify the expression when subtracting polynomials?

A: Yes, you can use algebraic properties to simplify the expression when subtracting polynomials. For example, you can use the distributive property to expand the expression and then combine like terms.

Q: How do I handle polynomials with multiple variables when subtracting polynomials?

A: When subtracting polynomials with multiple variables, you need to combine like terms and perform the subtraction operation for each variable separately.

Q: Can I use technology to help with subtracting polynomials?

A: Yes, you can use technology to help with subtracting polynomials. For example, you can use a graphing calculator or a computer algebra system to simplify the expression and perform the subtraction operation.

Q: How do I know when to use the distributive property when subtracting polynomials?

A: You should use the distributive property when subtracting polynomials when the expression contains a term that can be factored or expanded. This can help simplify the expression and make it easier to combine like terms.