Which Is Equivalent To The Expression Shown? Assume The Denominator Does Not Equal Zero.${ \frac{n^2+4n-12}{n-2} }$A. { N-6 $}$ B. { N-2 $}$ C. { N+2 $}$ D. { N+6 $}$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression n2+4n−12n−2\frac{n^2+4n-12}{n-2}. We will examine the different options provided and determine which one is equivalent to the given expression.

Understanding the Expression

The given expression is a rational expression, which means it is the ratio of two polynomials. The numerator is a quadratic expression, n2+4n−12n^2+4n-12, and the denominator is a linear expression, n−2n-2. To simplify this expression, we need to factor the numerator and cancel out any common factors between the numerator and the denominator.

Factoring the Numerator

The numerator, n2+4n−12n^2+4n-12, can be factored using the method of factoring quadratic expressions. We can start by finding two numbers whose product is −12-12 and whose sum is 44. These numbers are 66 and −2-2, so we can write the numerator as (n+6)(n−2)(n+6)(n-2).

Canceling Common Factors

Now that we have factored the numerator, we can see that it has a common factor with the denominator, which is (n−2)(n-2). We can cancel out this common factor by dividing both the numerator and the denominator by (n−2)(n-2). This leaves us with the simplified expression (n+6)(n+6).

Comparing with the Options

Now that we have simplified the expression, we can compare it with the options provided. The simplified expression is (n+6)(n+6), which matches option D. Therefore, the correct answer is D. (n+6)(n+6).

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By factoring the numerator and canceling out common factors, we can simplify complex expressions and determine which option is equivalent to the given expression. In this case, the simplified expression is (n+6)(n+6), which matches option D.

Frequently Asked Questions

Q: What is the process of simplifying algebraic expressions?

A: The process of simplifying algebraic expressions involves factoring the numerator and canceling out common factors between the numerator and the denominator.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the difference between a rational expression and a polynomial expression?

A: A rational expression is the ratio of two polynomials, while a polynomial expression is a single polynomial.

Final Answer

The final answer is D. (n+6)(n+6).

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given expression n2+4n−12n−2\frac{n^2+4n-12}{n-2}. In this article, we will provide a Q&A guide to help you better understand the concept of simplifying algebraic expressions.

Q&A Guide

Q: What is the purpose of simplifying algebraic expressions?

A: The purpose of simplifying algebraic expressions is to reduce complex expressions into simpler forms, making it easier to solve equations and inequalities.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and cancel out common factors between the numerator and the denominator.

Q: What is the difference between factoring and canceling?

A: Factoring involves breaking down an expression into its prime factors, while canceling involves eliminating common factors between the numerator and the denominator.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the difference between a rational expression and a polynomial expression?

A: A rational expression is the ratio of two polynomials, while a polynomial expression is a single polynomial.

Q: Can I simplify an expression with a zero denominator?

A: No, you cannot simplify an expression with a zero denominator. In fact, a zero denominator is undefined in mathematics.

Q: How do I know if an expression can be simplified?

A: An expression can be simplified if it has common factors between the numerator and the denominator.

Q: What is the final step in simplifying an expression?

A: The final step in simplifying an expression is to check if the expression can be further simplified by canceling out any remaining common factors.

Common Mistakes to Avoid

1. Not factoring the numerator

Not factoring the numerator can lead to an expression that cannot be simplified.

2. Not canceling common factors

Not canceling common factors can lead to an expression that is more complex than it needs to be.

3. Not checking for zero denominators

Not checking for zero denominators can lead to undefined expressions.

Tips and Tricks

1. Use factoring to simplify expressions

Factoring is a powerful tool for simplifying expressions.

2. Cancel common factors to simplify expressions

Canceling common factors can help simplify expressions and make them easier to work with.

3. Check for zero denominators before simplifying

Checking for zero denominators before simplifying can help avoid undefined expressions.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this Q&A guide, you can simplify complex expressions and make them easier to work with. Remember to factor the numerator, cancel common factors, and check for zero denominators to ensure that your expressions are simplified correctly.

Frequently Asked Questions

Q: What is the process of simplifying algebraic expressions?

A: The process of simplifying algebraic expressions involves factoring the numerator and canceling out common factors between the numerator and the denominator.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the difference between a rational expression and a polynomial expression?

A: A rational expression is the ratio of two polynomials, while a polynomial expression is a single polynomial.

Final Answer

The final answer is D. (n+6)(n+6).