Math Problem Statement

Simplify the following expression completely: (x^2 + 3x + 2) / (x^2 + 11x + 18)

Solution

The expression given is:

x2+3x+2x2+11x+18\frac{x^2 + 3x + 2}{x^2 + 11x + 18}

To simplify this, we will 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 xx).

Those two numbers are 11 and 22, so 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 x2+11x+18x^2 + 11x + 18. We need to find two numbers that multiply to 1818 and add to 1111.

Those two numbers are 22 and 99, so we can factor the denominator as: x2+11x+18=(x+2)(x+9)x^2 + 11x + 18 = (x + 2)(x + 9)

Step 3: Simplify the expression

Now that both the numerator and denominator are factored, we have:

(x+1)(x+2)(x+2)(x+9)\frac{(x + 1)(x + 2)}{(x + 2)(x + 9)}

We can cancel out the common factor (x+2)(x + 2) in both the numerator and denominator (as long as x2x \neq -2):

x+1x+9\frac{x + 1}{x + 9}

So, the simplified expression is:

x+1x+9,forx2\frac{x + 1}{x + 9}, \quad \text{for} \, x \neq -2

Would you like further details on any step? Here are 5 follow-up questions you might find useful:

  1. How do you factor quadratic expressions systematically?
  2. What does it mean to simplify a rational expression?
  3. Why is it important to note the restriction x2x \neq -2 in the simplification?
  4. Can this method be applied to higher-degree polynomials?
  5. How would the graph of this simplified function compare to the original?

Tip: Always check for restrictions on xx when simplifying rational expressions, as factors canceled out can indicate values that make the original expression undefined.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring Quadratic Expressions
Rational Expressions

Formulas

Factoring Quadratics
Simplification of Rational Expressions

Theorems

Zero Product Property
Cancellation of Common Factors

Suitable Grade Level

Grades 8-10