%2B7x%2B10)%2F(x%5E(2)%2B4x%2B4)*(x%5E(2)%2B3x%2B2)%2F(x%5E(2)%2B6x%2B5).jpeg "(x^(2)+7x+10)/(x^(2)+4x+4)*(x^(2)+3x+2)/(x^(2)+6x+5)")
Simplifying Rational Expressions: A Step-by-Step Guide
This article will guide you through the process of simplifying the rational expression:
(x^(2)+7x+10)/(x^(2)+4x+4) * (x^(2)+3x+2)/(x^(2)+6x+5)
1. Factoring the Expressions
The first step is to factor each of the quadratic expressions in the numerator and denominator. This will help us identify any common factors that can be canceled out.
Numerators:
- x^(2)+7x+10: This factors into (x+5)(x+2)
- x^(2)+3x+2: This factors into (x+2)(x+1)
Denominators:
- x^(2)+4x+4: This factors into (x+2)(x+2)
- x^(2)+6x+5: This factors into (x+5)(x+1)
2. Rewriting the Expression
Now, we can rewrite the entire expression with the factored terms:
[(x+5)(x+2)] / [(x+2)(x+2)] * [(x+2)(x+1)] / [(x+5)(x+1)]
3. Identifying Common Factors
Notice that we have several common factors in both the numerator and denominator:
- (x+2) appears twice in both the numerator and denominator.
- (x+5) and (x+1) also appear once in both the numerator and denominator.
4. Cancelling Common Factors
We can now cancel out these common factors:
[(x+5) * (x+2)] / [(x+2) * (x+2)] * [(x+2) * (x+1)] / [(x+5) * (x+1)] = 1 / (x+2)
5. Simplified Expression
The simplified expression is: 1 / (x+2)
Important Note: Remember that this simplified expression is valid only when the original expression is defined, meaning when the denominator is not equal to zero. In this case, x cannot be equal to -2, since it would make the denominator zero.