%2F(x%5E2-4%2Fx%5E2%2B5x%2B6).jpeg "(x^2-6x+8/3x-12)/(x^2-4/x^2+5x+6)")
Simplifying Complex Rational Expressions: A Step-by-Step Guide
This article will guide you through the process of simplifying the complex rational expression:
(x^2 - 6x + 8) / (3x - 12) / (x^2 - 4) / (x^2 + 5x + 6)
Step 1: Factor the expressions
- Numerator of the first fraction: (x^2 - 6x + 8) can be factored as (x - 4)(x - 2)
- Denominator of the first fraction: (3x - 12) can be factored as 3(x - 4)
- Numerator of the second fraction: (x^2 - 4) can be factored as (x + 2)(x - 2)
- Denominator of the second fraction: (x^2 + 5x + 6) can be factored as (x + 2)(x + 3)
Step 2: Rewrite the expression with factored terms
The original expression now becomes:
[(x - 4)(x - 2) / 3(x - 4)] / [(x + 2)(x - 2) / (x + 2)(x + 3)]
Step 3: Divide fractions
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we flip the second fraction and multiply:
[(x - 4)(x - 2) / 3(x - 4)] * [(x + 2)(x + 3) / (x + 2)(x - 2)]
Step 4: Cancel common factors
We can now cancel out the common factors in the numerator and denominator:
(x - 2) / 3 * (x + 3) / (x - 2)
Step 5: Simplify
Finally, we multiply the remaining terms:
(x + 3) / 3
Final Answer:
The simplified form of the complex rational expression is (x + 3) / 3.
Important Note: Remember to identify any restrictions on the domain of the expression. In this case, we need to exclude any values of x that make the denominator of the original expression equal to zero. Therefore, x cannot be equal to 4, -2, or -3.