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ICSE Class 10 Answered

Is (x+3) a factor of x^3+4x^2+x+6. What is the need to give zero of (x+3)
Asked by 21janhvi.verma | 29 May, 2018, 09:59: PM
answered-by-expert Expert Answer
begin mathsize 16px style To space check space left parenthesis straight x space plus space 3 right parenthesis space is space straight a space Factor space of space straight x cubed space plus space 4 straight x squared space plus space straight x space plus space 6 space.
We space will space equate space left parenthesis straight x space plus space 3 right parenthesis space equals space 0
For space cubic space equation space be space ax cubed plus space bx squared space plus cx space plus straight d space equals space 0 semicolon space straight a space not equal to space 0........... space space space space bold Standard bold space bold Form
The space cubic space polynomial space ax cubed plus space bx squared space plus cx space plus straight d space can space be space expressed space as space the space product space of space three space linear space factors comma space say space left parenthesis px space plus space straight q right parenthesis space comma left parenthesis rx space plus space straight s right parenthesis space and space left parenthesis tx space plus space straight u right parenthesis.
space where space straight p comma space straight q comma space straight r comma space straight s comma space straight t comma space straight u space space are space real space numbers space such space that space straight p space not equal to 0 space comma space straight r space not equal to space 0 space and space straight t not equal to 0 space. space Then comma space space ax cubed plus space bx squared space plus cx space plus straight d equals space 0 space is
space left parenthesis px space plus space straight q right parenthesis space left parenthesis rx space plus space straight s right parenthesis left parenthesis tx space plus space straight u right parenthesis space equals space 0 space space space..... left square bracket since comma space ax cubed plus space bx squared space plus cx space plus straight d space equals space left parenthesis px space plus space straight q right parenthesis space left parenthesis rx space plus space straight s right parenthesis left parenthesis tx space plus space straight u right parenthesis right square bracket
space px space plus space straight q space equals space 0 space or space rx space plus space straight s space equals space 0 space or space tx space plus space straight u space equals space 0
space Solving space these space linear space equations comma space
we space get space the space possible space roots space of space the space given space cubic space equation space as space straight x space equals space minus straight p divided by straight q space space or space space straight x space equals space minus straight s divided by straight r space space or space straight x space equals space minus straight u divided by straight t
left parenthesis straight x space plus space 3 right parenthesis space equals space 0
rightwards double arrow straight x space equals space minus 3
We space have space straight x cubed space plus space 4 straight x squared space plus space straight x space plus space 6 space equals space 0....... ax cubed plus space bx squared space plus cx space plus straight d space equals space 0 semicolon space straight a space not equal to space 0........... space space space space left parenthesis bold Standard bold space bold Form bold right parenthesis
put space straight x space equals negative 3 space space in space the space straight L. straight H. straight S space of space given space equation comma space we space get
straight L. straight H. straight S space equals space left parenthesis negative 3 right parenthesis cubed space plus space 4 left parenthesis negative 3 right parenthesis squared space plus space left parenthesis negative 3 right parenthesis plus space 6 space equals space 12 not equal to straight R. straight H. straight S
rightwards double arrow space left parenthesis straight x space plus space 3 right parenthesis space is space straight a space not space straight a space Factor space of space straight x cubed space plus space 4 straight x squared space plus space straight x space plus space 6.

end style
Answered by Yasmeen Khan | 30 May, 2018, 09:51: AM
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