The volume of a cuboid is polynomial p left parenthesis x right parenthesis equals 8 x cubed plus 12 x squared minus 2 x minus 3.Find possible expression for dimension of the cuboid.Verify the result by taking x=5units.

Asked by araima2001 | 13th Sep, 2014, 11:22: AM

Expert Answer:

C o n s i d e r space t h e space g i v e n space e x p r e s s i o n p open parentheses x close parentheses equals 8 x cubed plus 12 x squared minus 2 x minus 3 L e t space u s space t a k e space x equals 1 half p open parentheses 1 half close parentheses equals 8 open parentheses 1 half close parentheses cubed plus 12 open parentheses 1 half close parentheses squared minus 2 open parentheses 1 half close parentheses minus 3 rightwards double arrow p open parentheses 1 half close parentheses equals 8 cross times 1 over 8 plus 12 cross times 1 fourth minus 1 minus 3 rightwards double arrow p open parentheses 1 half close parentheses equals 1 plus 3 minus 1 minus 3 equals 0 T h u s comma space x equals 1 half comma space i s space a space r o o t space o f space t h e space c u b i c space e q u a t i o n. D i v i d e space t h e space e q u a t i o n space b y space t h e space f a c t o r space 2 x minus 1 T h u s comma space w e space h a v e comma fraction numerator 8 x cubed plus 12 x squared minus 2 x minus 3 over denominator 2 x minus 1 end fraction equals 4 x squared plus 8 x plus 3 L e t space u s space f a c t o r i z e comma space 4 x squared plus 8 x plus : 4 x squared plus 8 x plus 3 equals 4 x squared plus 6 x plus 2 x plus 3 equals 2 x open parentheses 2 x plus 3 close parentheses plus 1 open parentheses 2 x plus 3 close parentheses equals open parentheses 2 x plus 1 close parentheses open parentheses 2 x plus 3 close parentheses T h u s space t h e space d i m e n s i o n s space o f space t h e space c u b o i d space a r e : space 2 x plus 1 comma space 2 x plus 3 space a n d space 2 x minus 1 T h u s comma space t h e space v o l u m e space o f space t h e space c u b o i d space equals open parentheses 2 x plus 1 close parentheses open parentheses 2 x plus 3 close parentheses open parentheses 2 x minus 1 close parentheses N o w space s u b s t i t u e space x equals 5 space i n space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma V o l u m e equals open parentheses 2 cross times 5 plus 1 close parentheses open parentheses 2 cross times 5 plus 3 close parentheses open parentheses 2 cross times 5 minus 1 close parentheses equals 11 cross times 13 cross times 9 equals 1287 space c u b i c space u n i t s N o w space f r o m space t h e space e x p r e s s i o n space f o r space t h e space v o l u m e space o f space t h e space p o l y n o m i a l comma space w e space h a v V o l u m e equals 8 x cubed plus 12 x squared minus 2 x minus 3 S u b s t i t u t i n g space x equals 5 space i n space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma V o l u m e equals 8 cross times 5 cubed plus 12 cross times x squared minus 2 cross times 5 minus 3 rightwards double arrow V o l u m e equals 1000 plus 300 minus 10 minus 3 equals 1287 space c u b i c space u n i t s H e n c e space i t space h a s space b e e n space v e r i f i e d.

Answered by Vimala Ramamurthy | 14th Sep, 2014, 07:12: AM

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