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   show that (x+1) 8(2x-3)are factors of 2x^3-9x^2+x+12

Asked by seeni2005 | 14th May, 2016, 11:51: AM

Expert Answer:

C o n s i d e r space t h e space g i v e n space p o l y n o m i a l
2 x cubed minus 9 x squared plus x plus 12
W e space n e e d space t o space p r o v e space t h a t comma space x plus 1 comma space x minus 4 space a n d space 2 x minus 3 space a r e space f a c t o r s space o f space t h e space a b o v e space p o l y n o m i a l.
L e t space f open parentheses x close parentheses equals 2 x cubed minus 9 x squared plus x plus 12
C o n s i d e r space t h e space f a c t o r space left parenthesis x plus 1 right parenthesis
I f space left parenthesis x plus 1 right parenthesis space i s space a space f a c t o r comma space t h e n space b y space f a c t o r space t h e o r e m comma space x equals negative 1 space i s space a space z e r o space o f space t h e space p o l y n o m i a l.

L e t space u s space c o n s i d e r comma
f open parentheses negative 1 close parentheses equals 2 cross times open parentheses negative 1 close parentheses cubed minus 9 open parentheses negative 1 close parentheses squared plus open parentheses negative 1 close parentheses plus 12
space space space space space space space equals negative 2 minus 9 minus 1 plus 12
space space space space space space space equals 0
S i n c e space f open parentheses negative 1 close parentheses equals 0 comma space t h e n space open parentheses x plus 1 close parentheses space i s space a space f a c t o r space o f space t h e space g i v e n space p o l y n o m i a l.

S i m i l a r l y comma space l e t space u s space f i n d space f open parentheses 3 over 2 close parentheses colon
L e t space u s space c o n s i d e r comma
f open parentheses 3 over 2 close parentheses equals 2 cross times open parentheses 3 over 2 close parentheses cubed minus 9 open parentheses 3 over 2 close parentheses squared plus open parentheses 3 over 2 close parentheses plus 12
space space space space space space space equals 27 over 4 minus 81 over 4 plus open parentheses 3 over 2 close parentheses plus 12
space space space space space space space equals 27 over 4 minus 81 over 4 plus 6 over 4 plus 48 over 4
space space space space space space space equals fraction numerator 27 minus 81 plus 6 plus 48 over denominator 4 end fraction
space space space space space space space equals 0
S i n c e space f open parentheses 3 over 2 close parentheses equals 0 comma space t h e n space open parentheses 2 x minus 3 close parentheses space i s space a space f a c t o r space o f space t h e space g i v e n space p o l y n o m i a l.

S i m i l a r l y comma space l e t space u s space f i n d space f open parentheses 4 close parentheses colon
L e t space u s space c o n s i d e r comma
f open parentheses 4 close parentheses equals 2 cross times open parentheses 4 close parentheses cubed minus 9 open parentheses 4 close parentheses squared plus open parentheses 4 close parentheses plus 12
space space space space space space space equals 128 minus 144 plus 4 plus 12
space space space space space space equals 0
S i n c e space f open parentheses 4 close parentheses equals 0 comma space t h e n space open parentheses x minus 4 close parentheses space i s space a space f a c t o r space o f space t h e space g i v e n space p o l y n o m i a l.
T h u s comma space space open parentheses x plus 1 close parentheses comma space open parentheses 2 x minus 3 close parentheses comma space open parentheses x minus 4 close parentheses space a r e space t h e space f a c t o r s space o f space t h e space g i v e n space p o l y n o m i a l.

Answered by Vimala | 14th May, 2016, 02:48: PM