Simplify: -1 + ((x - b)( x - c ))/((a-b)(a-c)) +( (x - a)(x - b))/((c-a)(c-b)) +( (x - a)(x - c))/((b-a)(b-c))

The simplification is relatively lengthy so I will outline it: 
First off we set aside the -1 term and simplify the rational expression part. 

the lcd is 

(a-b)(a-c)(b-c) 

After some labor we arrive at these results: 
x^2 terms: 0 
ax terms: abx-acx 
bx terms: bcx-abx 
cx terms: acx-bcx 
bc terms: bc(b-c) 
ab terms: ab(a-b) 
ac terms: -ac(a-c) 

After collecting like terms we find that all x-terms add out (cancel each other to get a sum of 0) 

Leaving us with 

ab(a-b)-ac(a-c)+bc(b-c) 

which is equal to (a-b)(a-c)(b-c), the denominator! 

Therefore our rational expression simplifies to 1 so that the entire expression evaluates to 0 










:)