PARTIAL FRACTIONS

FIND THE PARTIAL FRACTION OF (x+14)/(x-4)(x+2)

(x+14)/(x-4)(x+2)=a/(x-4)+b/(x+2).......1.

(x+14)/(x-4)(x+2)=⟦a(x+2)+b(x-4)⟧(x-4)(x+2)

or

(x+14)=⟦a(x+2)+b(x-4)⟧

put 

x=-2

-2+14=⟦a(-2+2)+b(-2-4)⟧

12=b(-6)

b=-2

and

put

x=4

4+14=⟦a(4+2)+b(4-4)⟧

18=6a

a=3

hence

from 1.

(x+14)/(x-4)(x+2)=3/(x-4)-2/(x+2) 

Simplify (1/16x)+(1/8)) / ((1/4)+(1/8x))?

((1/16x)+(1/8)) / ((1/4)+(1/8x)) =y assume y=N/D

N=numerator=((1/16x)+(1/8))

and

D=(1/4)+(1/8x)

N=((1/16x)+(1/8))=

Now solve this

N=(1+2x)/16x

and

D=(1/4)+(1/8x)

D=(2x+1)/8x

Now y=N/D=(1+2x)/16x/(2x+1)/8x=8x/16x=1/2