Perform Long Division To Simplify The Expression { \frac{n^2 - 3n - 21}{n - 7}$}$.

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Introduction

In algebra, simplifying expressions is a crucial step in solving equations and inequalities. One of the most effective methods for simplifying expressions is long division. In this article, we will perform long division to simplify the expression {\frac{n^2 - 3n - 21}{n - 7}$}$. We will break down the process step by step, explaining each step in detail.

Understanding the Expression

The given expression is a rational expression, which is a fraction that contains variables in the numerator and denominator. The numerator is a quadratic expression, and the denominator is a linear expression. Our goal is to simplify this expression by dividing the numerator by the denominator.

Performing Long Division

To perform long division, we will follow these steps:

Step 1: Divide the Leading Term

The leading term of the numerator is n2n^2, and the leading term of the denominator is nn. To divide the leading term of the numerator by the leading term of the denominator, we will divide n2n^2 by nn, which gives us nn.

Step 2: Multiply and Subtract

We will multiply the entire denominator by the result from step 1, which is nn. This gives us n(n−7)=n2−7nn(n - 7) = n^2 - 7n. We will subtract this result from the numerator, which is n2−3n−21n^2 - 3n - 21. This gives us −7n−21-7n - 21.

Step 3: Bring Down the Next Term

The next term in the numerator is −21-21. We will bring this term down and write it below the result from step 2.

Step 4: Divide the Leading Term

The leading term of the result from step 3 is −7n-7n, and the leading term of the denominator is nn. To divide the leading term of the result from step 3 by the leading term of the denominator, we will divide −7n-7n by nn, which gives us −7-7.

Step 5: Multiply and Subtract

We will multiply the entire denominator by the result from step 4, which is −7-7. This gives us −7(n−7)=−7n+49-7(n - 7) = -7n + 49. We will subtract this result from the result from step 3, which is −7n−21-7n - 21. This gives us −70-70.

Step 6: Write the Final Result

The final result of the long division is n−7n - 7 with a remainder of −70-70. We can write this result as a fraction: {\frac{n^2 - 3n - 21}{n - 7} = n - 7 - \frac{70}{n - 7}$}$.

Conclusion

In this article, we performed long division to simplify the expression {\fracn^2 - 3n - 21}{n - 7}$}$. We broke down the process step by step, explaining each step in detail. The final result of the long division is n−7n - 7 with a remainder of −70-70. We can write this result as a fraction {\frac{n^2 - 3n - 21{n - 7} = n - 7 - \frac{70}{n - 7}$}$. This result can be further simplified by canceling out the common factor of n−7n - 7 in the numerator and denominator.

Simplifying the Result

Introduction

In our previous article, we performed long division to simplify the expression [\frac{n^2 - 3n - 21}{n - 7}\$}. We broke down the process step by step, explaining each step in detail. In this article, we will answer some frequently asked questions about simplifying algebraic expressions through long division.

Q: What is long division in algebra?

A: Long division in algebra is a method of dividing a polynomial by another polynomial. It is similar to long division in arithmetic, but it involves dividing variables and coefficients.

Q: How do I know when to use long division to simplify an expression?

A: You should use long division to simplify an expression when the numerator is a polynomial and the denominator is a linear expression. This is because long division is a method of dividing polynomials by linear expressions.

Q: What is the first step in performing long division?

A: The first step in performing long division is to divide the leading term of the numerator by the leading term of the denominator. This will give you the first term of the quotient.

Q: How do I multiply and subtract in long division?

A: To multiply and subtract in long division, you will multiply the entire denominator by the result from the previous step, and then subtract the result from the numerator.

Q: What is the remainder in long division?

A: The remainder in long division is the amount left over after you have divided the numerator by the denominator. It is usually a polynomial of lower degree than the numerator.

Q: Can I simplify the remainder in long division?

A: Yes, you can simplify the remainder in long division by factoring out any common factors. This will give you a simpler expression.

Q: How do I know when to stop simplifying an expression?

A: You should stop simplifying an expression when you have removed all the common factors and the remainder is a constant. This is because you cannot simplify a constant further.

Q: What are some common mistakes to avoid in long division?

A: Some common mistakes to avoid in long division include:

  • Not dividing the leading term of the numerator by the leading term of the denominator
  • Not multiplying and subtracting correctly
  • Not simplifying the remainder
  • Not stopping when the remainder is a constant

Conclusion

In this article, we answered some frequently asked questions about simplifying algebraic expressions through long division. We covered topics such as when to use long division, how to perform long division, and how to simplify the remainder. We also discussed some common mistakes to avoid in long division. By following these steps and avoiding these mistakes, you can simplify algebraic expressions with ease.

Additional Resources

For more information on simplifying algebraic expressions through long division, check out the following resources:

Practice Problems

Try these practice problems to test your skills in simplifying algebraic expressions through long division:

  • Simplify the expression {\frac{x^2 + 5x + 6}{x + 2}$}$
  • Simplify the expression {\frac{y^2 - 4y - 5}{y - 1}$}$
  • Simplify the expression {\frac{z^2 + 2z - 3}{z - 1}$}$

Answer Key

  • {\frac{x^2 + 5x + 6}{x + 2} = x + 3$}$
  • {\frac{y^2 - 4y - 5}{y - 1} = y + 5$}$
  • {\frac{z^2 + 2z - 3}{z - 1} = z + 3$}$