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
a^2+5a+4 = (a+1)(a+4)
a^2+3a+2 = (a+1)(a+2)
So the LCD = ......
A. To find the LCD (Least Common Denominator) for the given rational expressions, we need to factor the denominators.
The first rational expression has a denominator of a^2 + 5a + 4. To factor this quadratic expression, we need to find two numbers whose sum is 5 and product is 4. The factored form of the denominator is:
(a + 4)(a + 1).
The second rational expression has a denominator of a^2 + 3a + 2. To factor this quadratic expression, we need to find two numbers whose sum is 3 and product is 2. The factored form of the denominator is:
(a + 2)(a + 1).
B. Now, we can rewrite the rational expressions with the least common denominator.
The factored denominators are (a + 4)(a + 1) and (a + 2)(a + 1) respectively. To find the LCD, we need to include all the factors and take their product:
LCD = (a + 4)(a + 1)(a + 2).
Now, we can rewrite the rational expressions with the LCD:
For the first rational expression, we multiply the numerator and denominator by (a + 2) to get the LCD:
(5(a + 2))/[(a + 4)(a + 1)(a + 2)].
For the second rational expression, we multiply the numerator and denominator by (a + 4) to get the LCD:
(4a(a + 4))/[(a + 2)(a + 1)(a + 4)].
To find the least common denominator (LCD) for the given rational expressions, we need to factor the denominators and then identify the highest power of each factor that appears in any of the denominators.
A. Let's begin with the expression 5/a^2 + 5a + 4. We need to factor the denominator a^2 + 5a + 4 to find its factors.
The factors of a^2 + 5a + 4 are (a + 4)(a + 1).
Next, we need to consider the factors in terms of their highest power. In this case, both (a + 4) and (a + 1) have a highest power of 1.
So, the factors with their highest power are (a + 4) and (a + 1). Therefore, the LCD for the first rational expression is (a + 4)(a + 1).
B. Now, let's move on to the second rational expression 4a / (a^2 + 3a + 2). We need to factor the denominator a^2 + 3a + 2 to find its factors.
The factors of a^2 + 3a + 2 are (a + 2)(a + 1).
Similar to the first expression, both (a + 2) and (a + 1) have a highest power of 1.
So, the factors with their highest power are (a + 2) and (a + 1). Therefore, the LCD for the second rational expression is also (a + 2)(a + 1).
To rewrite the rational expressions with the least common denominator, we can multiply each expression by the missing factors from the other expression. Let's do it:
For the first expression, we multiply it by (a + 2) / (a + 2) since (a + 2) is missing from its denominator.
(5/a^2 + 5a + 4) * (a + 2) / (a + 2) = 5(a + 2) / [(a + 4)(a + 1)].
For the second expression, we multiply it by (a + 4) / (a + 4) since (a + 4) is missing from its denominator.
(4a / (a^2 + 3a + 2)) * (a + 4) / (a + 4) = 4a(a + 4) / [(a + 2)(a + 1)].
Now, both expressions have the same denominator, which is the least common denominator (LCD), and they are equivalent rational expressions.