simplify the following polynomial function

Algebra
Tutor: None Selected Time limit: 1 Day

Dec 2nd, 2014

write a,b,c always as cycle order as   ab , bc , ca  , a - b , b- c  , c - a 

P(x) = {- a(x-b)(x-c)/(a-b)(c-a)} +  {- b(x-a)(x-c)/(a-b)(b-c)}  +{- c(x-a)(x-b)/(c-a)(a-b)} , 

   take - sign  common from denominator to get same factors 

= {- a(b-c)(x-b)(x-c) - b(c-a)(x-a)(x-c)- c(a-b)(x-a)(x-b)} /(a-b)(b-c)(c-a)  ,  take lcm 

= {(ca-ab)(x^2-bx-cx+bc)+(ab-bc)(x^2-ax-cx+ca)+(bc-ca)(x^2-ax-bx+ab)}/(a-b)(b-c)(c-a) 

 multiply first 2 & last 2 terms in each term of numerator

={ x^2(ca-ab+ab-bc+bc-ca) +(ca-ab)(-bx-cx)+(ab-bc)(-ax-cx)+(bc-ca)(-ax-bx)+(ca-ab)bc+(ab-bc)ca)+

(bc-ca)ab)}/(a-b)(b-c)(c-a)  , multiply terms of x^2  , x  , constant terms

= {x^2(0) +x(ab^2-ac^2+bc^2-a^2b+a^2c-b^2c)+(abc^2-ab^2c+a^2bc-abc^2+ab^2c-a^2bc)}/(a-b)(b-c)(c-a)

= {0+x(a-b)(b-c)(c-a) +0}/(a-b)(b-c)(c-a) , ab^2-ac^2+bc^2-a^2b+a^2c-b^2c= (a-b)(b-c)(c-a)

= x

if not clear free feel to ask again .

Dec 3rd, 2014

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