Subtract These Polynomials:$\left(3x^2 + 2x + 4\right) - \left(x^2 + 2x + 1\right) =$A. $2x^2 + 4x + 3$B. $2x^2 + 5$C. $2x^2 + 3$D. $2x^2 + 4x + 5$

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Introduction

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. In this article, we will focus on subtracting polynomials, which is an essential operation in algebra. We will use the given problem to demonstrate the step-by-step process of subtracting polynomials.

What are Polynomials?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The general form of a polynomial is:

anxn+an−1xn−1+⋯+a1x+a0a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0

where an,an−1,…,a1,a0a_n, a_{n-1}, \ldots, a_1, a_0 are coefficients, 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 follow these steps:

  1. Write the polynomials to be subtracted: Write the two polynomials to be subtracted, one above the other, with the terms aligned.
  2. Combine like terms: Combine the like terms in each polynomial, which means combining the terms with the same variable and exponent.
  3. Subtract the coefficients: Subtract the coefficients of the like terms, making sure to change the sign of the coefficient of the second polynomial.
  4. Simplify the result: Simplify the result by combining any remaining like terms.

Example Problem

Let's use the given problem to demonstrate the step-by-step process of subtracting polynomials:

(3x2+2x+4)−(x2+2x+1)=?\left(3x^2 + 2x + 4\right) - \left(x^2 + 2x + 1\right) = ?

Step 1: Write the Polynomials to be Subtracted

Write the two polynomials to be subtracted, one above the other, with the terms aligned:

3x2+2x+4−x2−2x−1\begin{array}{r} 3x^2 + 2x + 4 \\ -x^2 - 2x - 1 \end{array}

Step 2: Combine Like Terms

Combine the like terms in each polynomial:

3x2−x2+2x−2x+4−1\begin{array}{r} 3x^2 - x^2 + 2x - 2x + 4 - 1 \end{array}

Step 3: Subtract the Coefficients

Subtract the coefficients of the like terms, making sure to change the sign of the coefficient of the second polynomial:

2x2+0x+3\begin{array}{r} 2x^2 + 0x + 3 \end{array}

Step 4: Simplify the Result

Simplify the result by combining any remaining like terms:

2x2+3\begin{array}{r} 2x^2 + 3 \end{array}

Conclusion

In this article, we demonstrated the step-by-step process of subtracting polynomials using the given problem. We wrote the polynomials to be subtracted, combined like terms, subtracted the coefficients, and simplified the result. The final answer is:

2x2+32x^2 + 3

This is the correct answer, which is option C.

Frequently Asked Questions

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

A: The difference between adding and subtracting polynomials is the sign of the second polynomial. When adding polynomials, we add the coefficients of the like terms. When subtracting polynomials, we subtract the coefficients of the like terms, making sure to change the sign of the coefficient of the second polynomial.

Q: How do I combine like terms in a polynomial?

A: To combine like terms in a polynomial, we add or subtract the coefficients of the like terms, making sure to keep the same variable and exponent.

Q: What is the final answer to the given problem?

A: The final answer to the given problem is 2x2+32x^2 + 3, which is option C.

Final Answer

The final answer is 2x2+32x^2 + 3.

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Introduction

In our previous article, we demonstrated the step-by-step process of subtracting polynomials. However, we understand that there may be some questions and doubts that readers may have. In this article, we will address some of the frequently asked questions about subtracting polynomials.

Q&A

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

A: The difference between adding and subtracting polynomials is the sign of the second polynomial. When adding polynomials, we add the coefficients of the like terms. When subtracting polynomials, we subtract the coefficients of the like terms, making sure to change the sign of the coefficient of the second polynomial.

Q: How do I combine like terms in a polynomial?

A: To combine like terms in a polynomial, we add or subtract the coefficients of the like terms, making sure to keep the same variable and exponent.

Q: What is the final answer to the given problem?

A: The final answer to the given problem is 2x2+32x^2 + 3, which is option C.

Q: Can I subtract polynomials with different variables?

A: No, you cannot subtract polynomials with different variables. Polynomials must have the same variable to be subtracted.

Q: Can I subtract polynomials with different exponents?

A: No, you cannot subtract polynomials with different exponents. Polynomials must have the same exponent to be subtracted.

Q: How do I simplify the result of subtracting polynomials?

A: To simplify the result of subtracting polynomials, we combine any remaining like terms.

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 check your work by hand to ensure accuracy.

Q: What is the importance of subtracting polynomials?

A: Subtracting polynomials is an essential operation in algebra, and it's used in a variety of applications, including solving equations, graphing functions, and finding the roots of a polynomial.

Q: Can I subtract polynomials with negative coefficients?

A: Yes, you can subtract polynomials with negative coefficients. When subtracting polynomials with negative coefficients, we change the sign of the coefficient of the second polynomial.

Q: Can I subtract polynomials with fractional coefficients?

A: Yes, you can subtract polynomials with fractional coefficients. When subtracting polynomials with fractional coefficients, we subtract the numerators and keep the same denominator.

Conclusion

In this article, we addressed some of the frequently asked questions about subtracting polynomials. We hope that this article has been helpful in clarifying any doubts or questions that readers may have had. Remember, subtracting polynomials is an essential operation in algebra, and it's used in a variety of applications.

Final Answer

The final answer is 2x2+32x^2 + 3.