A. Find the LCD for the given rational expression.
B. Rewrite them as equivalent rational expressions with the least common denominator.
5/a^2+5a+4,
4a/a^2+3a+2
https://www.jiskha.com/display.cgi?id=1507683563
To find the least common denominator (LCD) for the given rational expressions, we first need to factor the denominators of both expressions completely.
For the expression 5/a^2+5a+4, the denominator is a^2+5a+4. To factor this quadratic expression, we need to find two numbers whose product is 4 and sum is 5. It can be factored as (a+4)(a+1).
For the expression 4a/a^2+3a+2, the denominator is a^2+3a+2. Again, we need to find two numbers whose product is 2 and sum is 3. It can be factored as (a+2)(a+1).
Now, to find the LCD, we take the product of all the factors, including repeated factors, from both denominators. In this case, the LCD would be (a+4)(a+2)(a+1).
To rewrite the rational expressions with the least common denominator, we need to multiply each expression by the necessary factors to make their denominators equal to the LCD.
For the first expression, we multiply both the numerator and the denominator by (a+2)(a+1):
5/a^2+5a+4 * (a+2)(a+1)/(a+2)(a+1) = 5(a+2)(a+1) / (a+4)(a+2)(a+1)
For the second expression, we multiply both the numerator and the denominator by (a+4)(a+1):
4a/a^2+3a+2 * (a+4)(a+1)/(a+4)(a+1) = 4a(a+4)(a+1) / (a+4)(a+2)(a+1)
Thus, the equivalent rational expressions with the least common denominator are:
5(a+2)(a+1) / (a+4)(a+2)(a+1) and 4a(a+4)(a+1) / (a+4)(a+2)(a+1)