Subtract These Polynomials:$\left(3x^2 + 6x + 7\right) - \left(6x^2 - 5x - 7\right) =$A. $-3x^2 + 11x + 14$ B. $9x^2 + 11x + 14$ C. $9x^2 + X + 0$ D. $-3x^2 + X + 0$

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Introduction

Polynomial subtraction is a fundamental operation in algebra that involves subtracting one polynomial from another. It is an essential concept in mathematics, particularly in calculus and algebraic geometry. In this article, we will explore the process of subtracting polynomials, including the rules and steps involved.

What are Polynomials?

Before we dive into polynomial subtraction, let's briefly define 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 the form:

$$a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$$

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are coefficients, and $x$ is the variable.

Subtracting Polynomials

To subtract two polynomials, we need to follow the rules of polynomial subtraction. The rules are as follows:

1. Subtract the coefficients: Subtract the coefficients of the same degree from each other.
2. Combine like terms: Combine the like terms to simplify the expression.
3. Write the result: Write the resulting polynomial in the standard form.

Example: Subtracting Polynomials

Let's consider an example to illustrate the process of subtracting polynomials. We will subtract the polynomial $\left(3x^2 + 6x + 7\right)$ from the polynomial $\left(6x^2 - 5x - 7\right)$.

Step 1: Subtract the Coefficients

To subtract the coefficients, we need to subtract the coefficients of the same degree from each other.

$$\left(3x^2 + 6x + 7\right) - \left(6x^2 - 5x - 7\right)$$

Subtracting the coefficients of the same degree, we get:

$$\left(3x^2 - 6x^2\right) + \left(6x - \left(-5x\right)\right) + \left(7 - \left(-7\right)\right)$$

Step 2: Combine Like Terms

Now, we need to combine the like terms to simplify the expression.

$$\left(3x^2 - 6x^2\right) + \left(6x - \left(-5x\right)\right) + \left(7 - \left(-7\right)\right)$$

Combining the like terms, we get:

$$-3x^2 + 11x + 14$$

Step 3: Write the Result

The resulting polynomial is:

$$-3x^2 + 11x + 14$$

Conclusion

In this article, we have explored the process of subtracting polynomials. We have discussed the rules and steps involved in polynomial subtraction, including subtracting coefficients, combining like terms, and writing the result. We have also provided an example to illustrate the process of subtracting polynomials. By following these steps, you can easily subtract polynomials and simplify expressions.

Frequently Asked Questions

Q: What is polynomial subtraction?

A: Polynomial subtraction is the process of subtracting one polynomial from another.

Q: What are the rules of polynomial subtraction?

A: The rules of polynomial subtraction are:

1. Subtract the coefficients of the same degree from each other.
2. Combine the like terms to simplify the expression.
3. Write the resulting polynomial in the standard form.

Q: How do I subtract polynomials?

A: To subtract polynomials, follow the steps:

1. Subtract the coefficients of the same degree from each other.
2. Combine the like terms to simplify the expression.
3. Write the resulting polynomial in the standard form.

Final Answer

The final answer is $\boxed{-3x^2 + 11x + 14}$.

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Introduction

Polynomial subtraction is a fundamental operation in algebra that involves subtracting one polynomial from another. In this article, we will answer some of the most frequently asked questions about polynomial subtraction.

Q&A: Polynomial Subtraction

Q: What is polynomial subtraction?

A: Polynomial subtraction is the process of subtracting one polynomial from another. It involves subtracting the coefficients of the same degree from each other, combining like terms, and writing the resulting polynomial in the standard form.

Q: What are the rules of polynomial subtraction?

A: The rules of polynomial subtraction are:

1. Subtract the coefficients of the same degree from each other.
2. Combine the like terms to simplify the expression.
3. Write the resulting polynomial in the standard form.

Q: How do I subtract polynomials?

A: To subtract polynomials, follow the steps:

1. Subtract the coefficients of the same degree from each other.
2. Combine the like terms to simplify the expression.
3. Write the resulting polynomial in the standard form.

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

A: Polynomial addition and polynomial subtraction are two different operations. Polynomial addition involves adding polynomials, while polynomial subtraction involves subtracting polynomials.

Q: Can I subtract polynomials with different degrees?

A: Yes, you can subtract polynomials with different degrees. However, you need to make sure that the polynomials have the same variable and that the coefficients are of the same degree.

Q: How do I handle negative coefficients in polynomial subtraction?

A: When subtracting polynomials, you need to handle negative coefficients carefully. A negative coefficient is simply a coefficient with a negative sign. When subtracting polynomials, you need to subtract the coefficients of the same degree from each other, including negative coefficients.

Q: Can I simplify the expression after subtracting polynomials?

A: Yes, you can simplify the expression after subtracting polynomials. After subtracting the coefficients and combining like terms, you can simplify the expression by combining the like terms.

Q: What is the final answer in polynomial subtraction?

A: The final answer in polynomial subtraction is the resulting polynomial after subtracting the coefficients and combining like terms.

Example: Subtracting Polynomials with Negative Coefficients

Let's consider an example to illustrate the process of subtracting polynomials with negative coefficients. We will subtract the polynomial $\left(-3x^2 + 6x - 7\right)$ from the polynomial $\left(6x^2 - 5x + 7\right)$.

Step 1: Subtract the Coefficients

To subtract the coefficients, we need to subtract the coefficients of the same degree from each other.

$$\left(-3x^2 + 6x - 7\right) - \left(6x^2 - 5x + 7\right)$$

Subtracting the coefficients of the same degree, we get:

$$\left(-3x^2 - 6x^2\right) + \left(6x - \left(-5x\right)\right) + \left(-7 - \left(7\right)\right)$$

Step 2: Combine Like Terms

Now, we need to combine the like terms to simplify the expression.

$$\left(-3x^2 - 6x^2\right) + \left(6x - \left(-5x\right)\right) + \left(-7 - \left(7\right)\right)$$

Combining the like terms, we get:

$$-9x^2 + 11x - 14$$

Step 3: Write the Result

The resulting polynomial is:

$$-9x^2 + 11x - 14$$

Conclusion

In this article, we have answered some of the most frequently asked questions about polynomial subtraction. We have discussed the rules and steps involved in polynomial subtraction, including subtracting coefficients, combining like terms, and writing the resulting polynomial in the standard form. We have also provided an example to illustrate the process of subtracting polynomials with negative coefficients.

Final Answer

The final answer is $\boxed{-9x^2 + 11x - 14}$.