perform the indicated operation and simplify please help.
(v-5)/(v-6)-(v+1)/(v+6)+(v-18)/(v^2-36)
Notice that the denominator consists of
(v-6), (v+6), and (v^2-36)=(v+6)(v-6)
Therefore the common denominator is
(v²-36)=(v+6)(v-6)
The first fraction becomes:
(v-5)(v+6)
-------------
(v²-36)
The second fraction becomes:
(v+1)(v-6)
-------------
(v²-36)
And the third one stays at:
(v-18)
-------------
(v²-36)
Since we have a common denominator, we can expand the numberators and add to get:
7(v-6)
---------
(v+6)(v-6)
which after cancelling gives:
7/(v+6).
To perform the indicated operation and simplify the expression, we need to find a common denominator for all the fractions.
First, let's factor the denominator of the third fraction, v^2 - 36, which is a difference of squares:
v^2 - 36 = (v - 6)(v + 6)
Now, let's find the least common denominator (LCD) by considering the factors of each denominator:
LCD = (v - 6)(v + 6)(v + 6)(v - 6)
Next, we will rewrite each fraction with the common denominator:
(v - 5)/(v - 6) = [(v - 5)(v + 6)(v + 6)] / [(v - 6)(v + 6)(v + 6)(v - 6)]
(v + 1)/(v + 6) = [(v + 1)(v - 6)(v + 6)] / [(v - 6)(v + 6)(v + 6)(v - 6)]
(v - 18)/(v^2 - 36) = [(v - 18)] / [(v - 6)(v + 6)(v + 6)(v - 6)]
Now, we can combine the fractions by subtracting them:
[(v - 5)(v + 6)(v + 6)] / [(v - 6)(v + 6)(v + 6)(v - 6)] - [(v + 1)(v - 6)(v + 6)] / [(v - 6)(v + 6)(v + 6)(v - 6)] + [(v - 18)] / [(v - 6)(v + 6)(v + 6)(v - 6)]
Since we now have a common denominator, we can combine the numerators:
= [(v - 5)(v + 6)(v + 6) - (v + 1)(v - 6)(v + 6) + (v - 18)] / [(v - 6)(v + 6)(v + 6)(v - 6)]
Now, let's simplify the numerator:
(v - 5)(v + 6)(v + 6) - (v + 1)(v - 6)(v + 6) + (v - 18)
= (v^2 + v - 30)(v + 6) - (v^2 - 5v - 6)(v + 6) + v - 18
= (v^3 + 12v^2 + v - 180) - (v^3 + v^2 - 30v - 36) + v - 18
= v^3 + 12v^2 + v - 180 - v^3 - v^2 + 30v + 36 + v - 18
= 11v^2 + 32v - 162
Finally, the simplified expression is:
(11v^2 + 32v - 162) / [(v - 6)(v + 6)(v + 6)(v - 6)]