Math Problem Statement

Simplify the rationalize expression.

Solution

To simplify the given rational expression:

x2+3x+2x2x2\frac{x^2 + 3x + 2}{x^2 - x - 2}

we should first factor both the numerator and the denominator.

Step 1: Factor the numerator

The numerator is x2+3x+2x^2 + 3x + 2. We need to find two numbers that multiply to 22 (the constant term) and add to 33 (the coefficient of the middle term).

These numbers are 11 and 22. Therefore, we can factor the numerator as: x2+3x+2=(x+1)(x+2)x^2 + 3x + 2 = (x + 1)(x + 2)

Step 2: Factor the denominator

The denominator is x2x2x^2 - x - 2. We need to find two numbers that multiply to 2-2 (the constant term) and add to 1-1 (the coefficient of the middle term).

These numbers are 2-2 and 11. Therefore, we can factor the denominator as: x2x2=(x2)(x+1)x^2 - x - 2 = (x - 2)(x + 1)

Step 3: Simplify the expression

Now, substitute the factored forms back into the original expression: (x+1)(x+2)(x2)(x+1)\frac{(x + 1)(x + 2)}{(x - 2)(x + 1)}

We can cancel the common factor (x+1)(x + 1) in both the numerator and the denominator, as long as x1x \neq -1.

Thus, the simplified expression is: x+2x2\frac{x + 2}{x - 2}

Final Answer:

The simplified rational expression is x+2x2\frac{x + 2}{x - 2}, with the restriction that x1x \neq -1 (to prevent division by zero in the original expression).

Would you like further details or have any questions? Here are some related problems you might find interesting:

  1. Simplify x24x+4x24\frac{x^2 - 4x + 4}{x^2 - 4}.
  2. Factor and simplify 2x2+5x3x2+x6\frac{2x^2 + 5x - 3}{x^2 + x - 6}.
  3. Solve for xx in the equation x2+5x+6x2+2x3=1\frac{x^2 + 5x + 6}{x^2 + 2x - 3} = 1.
  4. Simplify the expression x3+3x2+3x+1x31\frac{x^3 + 3x^2 + 3x + 1}{x^3 - 1}.
  5. Determine the values of xx for which x29x2+3x4\frac{x^2 - 9}{x^2 + 3x - 4} is undefined.

Tip: When simplifying rational expressions, always check for common factors in the numerator and denominator, and don't forget to state any restrictions on the variable that arise from the original expression.

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Math Problem Analysis

Mathematical Concepts

Rational Expressions
Factoring

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12