*correction of previous post*
Find the LCD of the rational expressions in the list.
m-2/m^2+2m , -8/m^2+6m+8
(m-2)/m(m+2)
-8/(m+4)(m+2)
LCD = m(m+2)(m+4)
To find the least common denominator (LCD) of rational expressions, you need to factor the denominators and identify the highest power of each common factor.
Let's start by factoring the denominators of the two rational expressions:
For the first expression (m-2)/(m^2+2m):
The denominator is already in factored form, so we don't need to factorize it further.
For the second expression -8/(m^2+6m+8):
We need to factorize the denominator. The expression m^2+6m+8 does not factor nicely into linear factors, so we need to apply the quadratic formula.
Using the quadratic formula, we find that the roots of the quadratic equation m^2+6m+8 = 0 are:
m = (-6 ± √(6^2 - 4*1*8))/(2*1)
m = (-6 ± √(36 - 32))/(2)
m = (-6 ± √4)/(2)
m = (-6 ± 2)/(2)
Therefore, the quadratic factors as (m+2)(m+4).
Now that we have factored the denominators, let's identify the highest power of each common factor.
The factors in the first expression are (m-2) and (m+2).
The factors in the second expression are -8 and (m+2)(m+4).
The highest power of (m-2) is 1, the highest power of (m+2) is 1, and the highest power of (m+4) is 1.
To find the LCD, we multiply the highest powers of the common factors. In this case, the LCD is (m-2)(m+2)(m+4).
Therefore, the LCD of the rational expressions (m-2)/(m^2+2m) and -8/(m^2+6m+8) is (m-2)(m+2)(m+4).