Question

упростите выражение: 1/(х-1)(х-3)+1/(х-3)(х-5)+1/(х-5)(х-7)=

Answer 1

1/(х-1)(х-3)+1/(х-3)(х-5)+1/(х-5)(х-7)=

приведем к общему знаменателю -  (х-1)(х-3)(х-5)(х-7)

= [(x-5)*(x-7) + (x-1)*(x-7) + (x-1)*(x-3)]/(х-1)(х-3)(х-5)(х-7) =

= (x²-5x-7x+35 + x²-x-7x+7 + x²-x-3x+3)/(х-1)(х-3)(х-5)(х-7) =

= (3x²- 24x + 45)/(х-1)(х-3)(х-5)(х-7)=

= 3(x²- 8x + 15)/(х-1)(х-3)(х-5)(х-7)=

по теореме Виета

x²- 8x + 15 = 0 

х1=3

х2=5 

= 3(х-3)(х-5)/(х-1)(х-3)(х-5)(х-7)=

= 3/(х-1)(х-7)

Answer 2

числитель(х-5)(х-7)+(х-1)(х-7)+(х-1)(х-3)=

=x^2+35-12x+x^2-8x+7+x^2+3-4x=3x^2-24x+45=3(x^2-8x+15)=3(x-3)(x-5)

знаминатель (х-1)(х-3)(х-5)(х-7)

=3/(x-1)(x-7)