What Is The Difference?${ \frac{x}{x^2+3x+2} - \frac{1}{(x+2)(x+1)} }$A. { \frac{x-1}{6x+4}$}$B. { \frac{-1}{4x+2}$}$C. { \frac{1}{x+2}$}$D. { \frac{x-1}{x^2+3x+2}$}$

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What is the Difference? Simplifying Complex Rational Expressions

When dealing with complex rational expressions, it's essential to simplify them to make them more manageable and easier to work with. In this article, we'll explore the process of simplifying a complex rational expression, focusing on the given expression: xx2+3x+2−1(x+2)(x+1){\frac{x}{x^2+3x+2} - \frac{1}{(x+2)(x+1)}}

To simplify the given expression, we need to understand its components. The expression consists of two fractions: xx2+3x+2{\frac{x}{x^2+3x+2}} and 1(x+2)(x+1){\frac{1}{(x+2)(x+1)}}

The first fraction has a numerator of x and a denominator of x2+3x+2{x^2+3x+2}. The second fraction has a numerator of 1 and a denominator of (x+2)(x+1){(x+2)(x+1)}.

To simplify the expression, we need to find a common denominator for both fractions. The common denominator is the product of the denominators of both fractions: (x2+3x+2)(x+2)(x+1){(x^2+3x+2)(x+2)(x+1)}

Now, we can rewrite both fractions with the common denominator:

x(x+2)(x+1)(x2+3x+2)(x+2)(x+1)−(x+2)(x+1)(x2+3x+2)(x+2)(x+1){\frac{x(x+2)(x+1)}{(x^2+3x+2)(x+2)(x+1)} - \frac{(x+2)(x+1)}{(x^2+3x+2)(x+2)(x+1)}}

Now that both fractions have the same denominator, we can combine them by subtracting the numerators:

x(x+2)(x+1)−(x+2)(x+1)(x2+3x+2)(x+2)(x+1){\frac{x(x+2)(x+1) - (x+2)(x+1)}{(x^2+3x+2)(x+2)(x+1)}}

We can simplify the numerator by factoring out the common term (x+2)(x+1){(x+2)(x+1)}:

(x+2)(x+1)(x−(x+2))(x2+3x+2)(x+2)(x+1){\frac{(x+2)(x+1)(x - (x+2))}{(x^2+3x+2)(x+2)(x+1)}}

Now, we can simplify the numerator further by combining like terms:

(x+2)(x+1)(x−x−2)(x2+3x+2)(x+2)(x+1){\frac{(x+2)(x+1)(x - x - 2)}{(x^2+3x+2)(x+2)(x+1)}}

(x+2)(x+1)(−2)(x2+3x+2)(x+2)(x+1){\frac{(x+2)(x+1)(-2)}{(x^2+3x+2)(x+2)(x+1)}}

Now, we can simplify the expression by canceling out the common factors:

−2x2+3x+2{\frac{-2}{x^2+3x+2}}

In conclusion, the simplified expression is −2x2+3x+2{\frac{-2}{x^2+3x+2}}. This expression is equivalent to the original expression, but it's much simpler and easier to work with.

The correct answer is D. x−1x2+3x+2{\frac{x-1}{x^2+3x+2}}
What is the Difference? Simplifying Complex Rational Expressions - Q&A

In our previous article, we explored the process of simplifying a complex rational expression. We walked through the steps of finding a common denominator, combining the fractions, and simplifying the numerator. In this article, we'll answer some frequently asked questions about simplifying complex rational expressions.

A: A common denominator is the product of the denominators of two or more fractions. It's the denominator that both fractions have in common.

A: To find a common denominator, you need to multiply the denominators of both fractions together. For example, if you have two fractions with denominators of x and x+1, the common denominator would be x(x+1).

A: If the denominators are not factorable, you can still find a common denominator by multiplying the denominators together. For example, if you have two fractions with denominators of x^2+3x+2 and x+2, the common denominator would be (x^2+3x+2)(x+2).

A: Yes, you can simplify a complex rational expression by canceling out common factors. This is a common technique used to simplify expressions.

A: If you have a fraction with a numerator and denominator that are both polynomials, you can simplify the expression by factoring the numerator and denominator. Then, you can cancel out any common factors.

A: Yes, you can use a calculator to simplify a complex rational expression. However, it's always a good idea to check your work by simplifying the expression manually.

A: If you get stuck while simplifying a complex rational expression, try breaking it down into smaller steps. You can also try using a different method or technique to simplify the expression.

A: Yes, here are a few tips for simplifying complex rational expressions:

  • Make sure to find a common denominator before combining the fractions.
  • Factor the numerator and denominator to simplify the expression.
  • Cancel out any common factors to simplify the expression.
  • Use a calculator to check your work.

In conclusion, simplifying complex rational expressions can be a challenging task, but with practice and patience, you can master it. Remember to find a common denominator, factor the numerator and denominator, and cancel out any common factors to simplify the expression. If you get stuck, try breaking it down into smaller steps or using a different method or technique.

Here are a few common mistakes to avoid when simplifying complex rational expressions:

  • Not finding a common denominator before combining the fractions.
  • Not factoring the numerator and denominator to simplify the expression.
  • Not canceling out any common factors to simplify the expression.
  • Not using a calculator to check your work.

Here are a few final tips for simplifying complex rational expressions:

  • Practice, practice, practice! The more you practice, the more comfortable you'll become with simplifying complex rational expressions.
  • Use a calculator to check your work and make sure you're getting the correct answer.
  • Don't be afraid to ask for help if you get stuck.
  • Take your time and break down the expression into smaller steps if necessary.

The correct answer is D. x−1x2+3x+2{\frac{x-1}{x^2+3x+2}}