What are the steps to find zeroes of all types of polynomials?Thanks

Asked by aayush.rattan | 4th May, 2016, 11:02: PM

Expert Answer:

C o n s i d e r space t h e space p o l y n o m i a l space P open parentheses x close parentheses equals 2 x to the power of 5 plus 3 x to the power of 4 minus 30 x cubed minus 57 x squared minus 2 x plus 24.

W e space n e e d space t o space d o space t r i a l space a n d space e r r o r space m e t h o d space f i r s t space t o space s o l v e space f o r space z e r o e s.
W h e n space x equals negative 1 comma space P open parentheses x close parentheses equals 2 open parentheses negative 1 close parentheses to the power of 5 plus 3 open parentheses negative 1 close parentheses to the power of 4 minus 30 open parentheses negative 1 close parentheses cubed minus 57 open parentheses negative 1 close parentheses squared minus 2 open parentheses negative 1 close parentheses plus 24 equals negative 2 plus 3 plus 30 minus 57 plus 2 plus 24 equals 0
T h u s comma space x equals negative 1 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.

L e t space u s space f i n d space t h e space q u o t i e n t space w h e n space 2 x to the power of 5 plus 3 x to the power of 4 minus 30 x cubed minus 57 x squared minus 2 x plus 24 space i s space d i v i d e d space b y space open parentheses x plus 1 close parentheses
 
-1 2 3 -30 -57 -2 24
  0 -2 -1 31 26 -24
  2 1 -31 -26 24 0
 
T h u s comma space i t space i s space e n o u g h space i f space w e space f i n d space t h e space r o o t s space o f space t h e space p o l y n o m i a l comma space 2 x to the power of 4 plus x cubed minus 31 x squared minus 26 plus 24
N o w space t h e space d e g r e e space o f space t h e space p o l y n o m i a l space i s space 4.
T h u s comma space t h e r e space a r e space 4 space r o o t s space w e space n e e d space t o space f i n d.
 
L e t space u s space f i n d space t h e space r o o t s space b y space r a t i o n a l space r o o t space t h e o r e m.
R a t i o n a l space r o o t space t h e o r e m colon
C o n s i d e r space a space p o l y n o m i a l space e q u a t i o n
P open parentheses x close parentheses equals a subscript n x to the power of n plus a subscript n minus 1 end subscript x to the power of n minus 1 end exponent plus... plus a subscript 0 space equals 0 space w i t h space i n t e g e r space c o e f f i c i e n t s.
I f space a subscript 0 space a n d space a subscript n space a r e space n o n minus z e r o comma space t h e n space e a c h space r a t i o n a l space s o l u t i o n space x comma space w h e n space w r i t t e n space a s space a space f r a c t i o n
x equals p over q space i n space l o w e s t space t e r m s comma space s a t i s f i e s
1. space p space i s space a n space i n t e g e r space f a c t o r space o f space t h e space c o n s t a n t space t e r m space a subscript 0
2. space q space i s space a n space i n t e g e r space f a c t o r space o f space t h e space l e a d i n g space c o e f f i c i e n t space a subscript n

T h u s comma space b y space r a t i o n a l space r o o t s space t h e o r e m comma space t h e space p o s s i b l e space z e r o e s space o f space t h e space r e d u c e d space p o y n o m i a l space 2 x to the power of 4 plus x cubed minus 31 x squared minus 26 x plus 24
a r e colon space plus-or-minus fraction numerator 1 comma 2 comma 3 comma 4 comma 6 comma 8 comma 12 comma 24 over denominator 1 comma 2 end fraction equals plus-or-minus 1 half comma space 1 comma space 1 third comma 2 comma 3 comma 4 comma 6 comma 8 comma 12 comma 24

L e t space u s space t r y space t h e space r o o t space x equals 2
f left parenthesis x right parenthesis equals 2 open parentheses 2 close parentheses to the power of 4 plus open parentheses 2 close parentheses cubed minus 31 open parentheses 2 close parentheses squared minus 26 open parentheses 2 close parentheses plus 24
space space space space space space space space equals 32 plus 8 minus 124 minus 52 plus 24
space space space space space space space space equals 64 minus 176
space space space space space space space space equals negative 112
space space space space space space space space not equal to 0
T h u s comma space x equals 2 space i s space n o t space a space z e r o space o f space t h e space r e d u c e d space p o l y n o m i a l.
L e t space u s space t r y space f o r space x equals 4

Answered by Vimala | 5th May, 2016, 10:13: AM