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  If (x+a)(x+b)(x+c) ≡ x³-10x² + 45x -15 find a+b+c, 1/a+1/b+1/c and 

a²+b²+c².

Asked by nagul2793 15th October 2017, 3:30 PM
Answered by Expert
Answer:
begin mathsize 16px style left parenthesis straight x plus straight a right parenthesis left parenthesis straight x plus straight b right parenthesis left parenthesis straight x plus straight c right parenthesis equals straight x cubed space minus space 10 straight x squared space plus space 45 straight x space minus space 15
open parentheses straight x squared plus ax plus bx plus ab close parentheses left parenthesis straight x plus straight c right parenthesis equals straight x cubed space minus space 10 straight x squared space plus space 45 straight x space minus space 15
straight x cubed plus straight x squared straight c plus ax squared plus axc space plus space bx squared plus bxc plus plus abx plus abc equals straight x cubed space minus space 10 straight x squared space plus space 45 straight x space minus space 15
straight x cubed space plus space open parentheses straight a plus straight b plus straight c close parentheses straight x squared space plus space open parentheses ab plus bc plus ca close parentheses straight x plus abc space equals space straight x cubed space minus space 10 straight x squared space plus space 45 straight x space minus space 15
Comparing space coefficients space of space both space sides
straight a plus straight b plus straight c space equals space minus 10........ left parenthesis straight i right parenthesis
ab plus bc plus ca space equals space 45...... left parenthesis ii right parenthesis
abc space equals space minus 15.............. left parenthesis iii right parenthesis
Dividing space equation space left parenthesis straight i right parenthesis space by space left parenthesis ii right parenthesis
fraction numerator ab plus bc plus ca over denominator abc end fraction space equals fraction numerator space 45 over denominator negative 15 end fraction
ab over abc plus bc over abc plus ca over abc space equals space minus 3
1 over straight c plus 1 over straight b plus 1 over straight a equals negative 3
To space find space straight a squared plus straight b squared plus straight c squared comma
open parentheses straight a plus straight b plus straight c close parentheses squared space equals space space straight a squared plus straight b squared plus straight c squared plus 2 ab plus 2 bc plus 2 ca
open parentheses negative 10 close parentheses squared space equals space straight a squared space plus straight b squared plus straight c squared space plus space 2 open parentheses ab plus bc plus ca close parentheses
100 space equals space straight a squared space plus straight b squared plus straight c squared space plus space 2 cross times 45 space space space space space space space space space space space space space space space space space space space space space space
straight a squared space plus straight b squared plus straight c squared space equals space 100 space minus space 90
straight a squared space plus straight b squared plus straight c squared space equals space 10

Answers colon
straight a plus straight b plus straight c space equals space minus 10
1 over straight c plus 1 over straight b plus 1 over straight a equals negative 3
straight a squared space plus straight b squared plus straight c squared space equals space 10

end style
Answered by Expert 4th December 2017, 3:22 PM
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