Simplify By Using Long Division: X 2 + 2 X − 6 X − 2 \frac{x^2 + 2x - 6}{x - 2} X − 2 X 2 + 2 X − 6 ​ A) X + 2 − 4 X − 2 X + 2 - \frac{4}{x - 2} X + 2 − X − 2 4 ​ B) X + 4 − 2 X − 2 X + 4 - \frac{2}{x - 2} X + 4 − X − 2 2 ​ C) X + 4 + 2 X − 2 X + 4 + \frac{2}{x - 2} X + 4 + X − 2 2 ​ D) X + 2 + 4 X − 2 X + 2 + \frac{4}{x - 2} X + 2 + X − 2 4 ​

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Introduction

Long division is a mathematical technique used to simplify complex fractions by dividing one polynomial by another. In this article, we will use long division to simplify the given fraction $\frac{x^2 + 2x - 6}{x - 2}$. We will explore the different steps involved in the long division process and provide a clear explanation of each step.

Step 1: Set up the long division

To begin the long division process, we need to set up the fraction $\frac{x^2 + 2x - 6}{x - 2}$ in a way that allows us to perform the division. We can do this by writing the dividend (the numerator) and the divisor (the denominator) in a long division format.

Step 2: Divide the leading term of the dividend by the leading term of the divisor

The leading term of the dividend is $x^2$, and the leading term of the divisor is $x$. To divide $x^2$ by $x$, we get $x$. This is the first term of the quotient.

Step 3: Multiply the divisor by the first term of the quotient

We multiply the divisor $x - 2$ by the first term of the quotient $x$, which gives us $x^2 - 2x$.

Step 4: Subtract the product from the dividend

We subtract the product $x^2 - 2x$ from the dividend $x^2 + 2x - 6$, which gives us $4x - 6$.

Step 5: Bring down the next term of the dividend

Since the dividend is a quadratic expression, we need to bring down the next term, which is $-6$.

Step 6: Divide the leading term of the new dividend by the leading term of the divisor

The leading term of the new dividend is $4x$, and the leading term of the divisor is $x$. To divide $4x$ by $x$, we get $4$. This is the next term of the quotient.

Step 7: Multiply the divisor by the next term of the quotient

We multiply the divisor $x - 2$ by the next term of the quotient $4$, which gives us $4x - 8$.

Step 8: Subtract the product from the new dividend

We subtract the product $4x - 8$ from the new dividend $4x - 6$, which gives us $2$.

Step 9: Write the final quotient and remainder

The final quotient is $x + 4$, and the remainder is $2$.

Conclusion

Using long division, we have simplified the given fraction $\frac{x^2 + 2x - 6}{x - 2}$ to $x + 4 - \frac{2}{x - 2}$. This is the correct answer.

Comparison with the given options

Let's compare our answer with the given options:

• Option A: $x + 2 - \frac{4}{x - 2}$
• Option B: $x + 4 - \frac{2}{x - 2}$
• Option C: $x + 4 + \frac{2}{x - 2}$
• Option D: $x + 2 + \frac{4}{x - 2}$

Our answer matches option B, which is $x + 4 - \frac{2}{x - 2}$.

Final Answer

The final answer is $\boxed{x + 4 - \frac{2}{x - 2}}$.

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Introduction

In our previous article, we used long division to simplify the given fraction $\frac{x^2 + 2x - 6}{x - 2}$. We obtained the simplified form as $x + 4 - \frac{2}{x - 2}$. In this article, we will answer some frequently asked questions related to the long division process and provide additional insights into the topic.

Q&A

Q1: What is long division in mathematics?

A1: Long division is a mathematical technique used to simplify complex fractions by dividing one polynomial by another. It involves dividing the dividend (the numerator) by the divisor (the denominator) to obtain a quotient and a remainder.

Q2: How do I set up the long division?

A2: To set up the long division, you need to write the dividend (the numerator) and the divisor (the denominator) in a long division format. The dividend should be written on top of the line, and the divisor should be written below the line.

Q3: What is the first step in the long division process?

A3: The first step in the long division process is to divide the leading term of the dividend by the leading term of the divisor. This will give you the first term of the quotient.

Q4: How do I multiply the divisor by the first term of the quotient?

A4: To multiply the divisor by the first term of the quotient, you need to multiply each term of the divisor by the first term of the quotient. For example, if the divisor is $x - 2$ and the first term of the quotient is $x$, you would multiply $x$ by $x$ to get $x^2$, and multiply $x$ by $-2$ to get $-2x$.

Q5: What is the remainder in the long division process?

A5: The remainder in the long division process is the amount left over after you have divided the dividend by the divisor. It is usually a polynomial expression.

Q6: How do I write the final quotient and remainder?

A6: To write the final quotient and remainder, you need to combine the terms of the quotient and the remainder. The quotient is the result of the division, and the remainder is the amount left over.

Q7: Can I use long division to simplify any type of fraction?

A7: No, you cannot use long division to simplify any type of fraction. Long division is used to simplify fractions where the numerator and denominator are polynomials. If the numerator and denominator are not polynomials, you may need to use a different method to simplify the fraction.

Q8: How do I check my answer?

A8: To check your answer, you can multiply the quotient by the divisor and add the remainder to the product. If the result is equal to the original dividend, then your answer is correct.

Conclusion

In this article, we have answered some frequently asked questions related to the long division process and provided additional insights into the topic. We hope that this article has been helpful in clarifying any doubts you may have had about long division.

Final Answer

The final answer is $\boxed{x + 4 - \frac{2}{x - 2}}$.