Which Polynomial Represents The Difference Below?$\[ \begin{array}{r} 5x^2 + 9x + 3 \\ - \quad (6x^2 - 3x) \\ \hline \end{array} \\]A. $5x^2 + 3x + 3$ B. $-x^2 + 12x + 3$ C. $5x^2 + 6x + 3$ D. $-x^2 + 6x +

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In algebra, polynomial subtraction is a fundamental operation that involves finding the difference between two polynomials. This process is essential in various mathematical applications, including solving equations, graphing functions, and simplifying expressions. In this article, we will explore the concept of polynomial subtraction and provide a step-by-step guide on how to perform it.

Understanding Polynomials

Before we dive into polynomial subtraction, it's essential to understand what polynomials are. A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be written in various forms, including:

  • Monomials: A single term with a variable and a coefficient, such as 3x or 2y.
  • Binomials: Two terms combined using addition or subtraction, such as x + 3 or 2x - 4.
  • Polynomials: A sum of monomials, such as 2x^2 + 3x - 4 or x^3 - 2x^2 + x + 1.

The Subtraction Process

Now that we have a basic understanding of polynomials, let's move on to the subtraction process. When subtracting polynomials, we need to follow the same rules as when adding polynomials, but with a twist. When subtracting a term, we change the sign of the term and then combine it with the other terms.

Step 1: Write the Polynomials

The first step in polynomial subtraction is to write the two polynomials side by side, with the terms aligned. In the given problem, we have:

{ \begin{array}{r} 5x^2 + 9x + 3 \\ - \quad (6x^2 - 3x) \\ \hline \end{array} \}

Step 2: Subtract the Terms

Next, we need to subtract the terms of the second polynomial from the first polynomial. We start by subtracting the term with the highest degree (i.e., the term with the largest exponent). In this case, we have:

  • 5x2−6x2=−x25x^2 - 6x^2 = -x^2
  • 9x+3x=12x9x + 3x = 12x
  • 3−0=33 - 0 = 3

Step 3: Combine the Terms

Now that we have subtracted the terms, we need to combine them to get the final result. In this case, we have:

−x2+12x+3-x^2 + 12x + 3

Conclusion

In conclusion, polynomial subtraction is a fundamental operation in algebra that involves finding the difference between two polynomials. By following the steps outlined in this article, you can perform polynomial subtraction with ease. Remember to write the polynomials side by side, subtract the terms, and combine the results to get the final answer.

Answer

The correct answer is:

  • B. −x2+12x+3-x^2 + 12x + 3

Tips and Tricks

Here are some tips and tricks to help you perform polynomial subtraction like a pro:

  • Use the distributive property: When subtracting a term, use the distributive property to change the sign of the term and then combine it with the other terms.
  • Combine like terms: When combining the terms, make sure to combine like terms (i.e., terms with the same variable and exponent).
  • Check your work: Finally, check your work by plugging in some values for the variable to make sure the result is correct.

In our previous article, we explored the concept of polynomial subtraction and provided a step-by-step guide on how to perform it. However, we know that practice makes perfect, and there's no better way to practice than by answering questions. In this article, we'll provide a Q&A section on polynomial subtraction, covering various scenarios and examples.

Q: What is the difference between polynomial addition and polynomial subtraction?

A: Polynomial addition and polynomial subtraction are both operations that involve combining or subtracting polynomials. However, the key difference between the two is that addition involves combining like terms, while subtraction involves changing the sign of the terms and then combining them.

Q: How do I perform polynomial subtraction when the polynomials have different degrees?

A: When the polynomials have different degrees, you need to perform polynomial subtraction by subtracting the terms with the highest degree first. For example, if you have:

{ \begin{array}{r} x^2 + 3x + 1 \\ - \quad (2x^3 - 4x^2 + 3x) \\ \hline \end{array} \}

You would first subtract the term with the highest degree (i.e., the term with the largest exponent), which is 2x32x^3. Then, you would subtract the remaining terms.

Q: What if I have a polynomial with a negative coefficient? How do I perform polynomial subtraction?

A: When you have a polynomial with a negative coefficient, you need to change the sign of the term when subtracting it. For example, if you have:

{ \begin{array}{r} x^2 + 3x + 1 \\ - \quad (-2x^2 + 4x - 3) \\ \hline \end{array} \}

You would first change the sign of the term −2x2-2x^2 to 2x22x^2, and then subtract the remaining terms.

Q: Can I perform polynomial subtraction with polynomials that have different variables?

A: Yes, you can perform polynomial subtraction with polynomials that have different variables. However, you need to make sure that the variables are the same or that the polynomials can be rewritten in terms of the same variable. For example, if you have:

{ \begin{array}{r} x^2 + 3x + 1 \\ - \quad (y^2 - 4y + 3) \\ \hline \end{array} \}

You would need to rewrite the polynomial y2−4y+3y^2 - 4y + 3 in terms of the variable xx.

Q: How do I check my work when performing polynomial subtraction?

A: To check your work when performing polynomial subtraction, you can plug in some values for the variable and make sure the result is correct. For example, if you have:

{ \begin{array}{r} x^2 + 3x + 1 \\ - \quad (2x^2 - 4x + 3) \\ \hline \end{array} \}

You can plug in x=1x = 1 and make sure the result is correct.

Q: What are some common mistakes to avoid when performing polynomial subtraction?

A: Some common mistakes to avoid when performing polynomial subtraction include:

  • Not changing the sign of the terms: When subtracting a term, make sure to change the sign of the term.
  • Not combining like terms: When combining the terms, make sure to combine like terms (i.e., terms with the same variable and exponent).
  • Not checking your work: Finally, make sure to check your work by plugging in some values for the variable to make sure the result is correct.

By following these tips and avoiding common mistakes, you can perform polynomial subtraction with ease and accuracy.