polynomials-maths fun-by ppant85

8/9/2019 Polynomials-Maths Fun-By ppant85

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WORK B ASED ON :- P OLYNOMIALSINTRODUCTION

DEGREE OF A POLYNOMIAL IN ONE VARIABLEDEGREE OF A POLYNOMIAL IN TWO VARIABLEPOLYNOMIALS IN ONE VARIABLE(DEFINATION)

EXAMPLESREMAINDER THEOREMEXTRA QUESTIONS ON REMAINDER THEOREM

FACTOR THEOREMALGEBRAIC IDENTITIES

QUESTIONS ON POLYNOMIALSPOINTS TO REMEMBER

Made !"# PRADEEP PANT($$a%&' *+a,-./0+)


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INTRODUCTION:

I%&10d2/&,0% A% a-*e 1a,/ e3$1e44,0% ,%56,/6 7a1,a -e4 ,%70-7ed 6a7e 0%-! %0%#%e*a&,7e ,%&e*1a- $05e14 ,4 /a--ed a$0-!%0+,a-. E.*.# (a) 839:;38


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DEGREE OF A POLYNOMIAL IN ONE VARIABLE :

De*1ee 0 a P0-!%0+,a- ,% 0%e7a1,a -e. W6a& ,4 de*1ee 0 &6e0--05,%* ,%0+,a- T6e a%45e1 ,4 8.38 < 9 ,4 a $0-!%0+,a- ,% 3 0de*1ee 8. I% /a4e 0 a $0-!%0+,a- ,%

0%e 7a1,a -e &6e 6,*6e4& $05e1 0&6e 7a1,a -e ,4 /a--ed &6e de*1ee 0$0-!%0+,a-.


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POLYNOMIALS IN ONE VARIABLE:

P0-!%0+,a-4 ,% 0%e 7a1,a -e A$0-!%0+,a- ,4 a +0%0+,a- 01 a 42+ 0+0%0+,a-4. Ea/6 +0%0+,a- ,% a$0-!%0+,a- ,4 a &e1+ 0 &6e $0-!%0+,a-.

T6e %2+ e1 a/&01 0 a &e1+ ,4 /a--ed&6e /0e /,e%&. T6e /0e /,e%& 0 &6e14& &e1+ ,% a $0-!%0+,a- ,4 &6e -ead

/0e /,e%&.A $0-!%0+,a- 5,&6 &50 &e1+4 ,4 /a--ed a,%0+,a-. A $0-!%0+,a- 5,&6 &61ee &e1+

,4 /a--ed a &1,%0+,a-.


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POLYNOMIALS IN ONE VARIABLE:

P0-!%0+,a-4 ,% 0%e 7a1,a -e T6e de*1ee 0 a$0-!%0+,a- ,% 0%e 7a1,a -e ,4 &6e -a1*e4&e3$0%e%& 0 &6a& 7a1,a -e. = ,4 a /0%4&a%& 6a4 %0 7a1,a -e. I& ,4 a de*1ee $0-!%0+,a-. 3


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REMAINDER T*EOREM :

Le& (3) e a $0-!%0+,a- 0 de*1ee % ? a%d -e& a e a%! 1ea-%2+ e1. W6e% (3) ,4 d,7,ded ! (3#a) &6e% &6e 1e+a,%de1 ,4(a)

. PROOF S2$$04e 56e% (3) ,4 d,7,ded ! (3#a) &6e 20&,e%& ,4*(3) a%d &6e 1e+a,%de1 ,4 1(3).

T6e% de*1ee 1(3) de*1ee (3#a) de*1ee 1(3) ? &6e1e 01e de*1ee (3#a) ?de*1ee 1(3) 1(3) ,4 /0%4&a%& e 2a- &0 1 (4a!)

T624 56e% (3) ,4 d,7,ded ! (3#a) &6e% &6e 20&,e%& ,4 *@3)a%d &6e 1e+a,%de1 ,4 1. T6e1e 01e (3) (3#a) *(3) < 1 (,) P2&&,%* 3 a ,% (,) 5e *e& 1

(a) . T624 56e% (3) ,4 d,7,ded ! (3#a) &6e% &6e 1e+a,%de1 ,4 (a).


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EXTRA +UESTION ON REMAINDERT*EOREM :

Q2e4&,0%4 0% Re+a,%de1 T6e01e+ Q.) F,%d&6e 1e+a,%de1 56e% &6e $0-!%0+,a- (3) 3; < 839 : 938 < 3 : ? ,4 d,7,ded ! (3#8).

A.) 3#8 3 8 B! 1e+a,%de1 &6e01e+5e %05 &6a& 56e% (3) ,4 d,7,ded ! (3#8) &6e 1e+a,%de1 ,4 3(8).

N05 (8) (8; < 8 89 : 9 88 < 8#?) (?= < ?= : ?8 < 8 : ?) 8?. He%/e &6e1e 2,1ed 1e+a,%de1 ,4 8?.


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FACTOR T*EOREM :

Fa/&01 T6e01e+ Le& (3) e a $0-!%0+,a- 0de*1ee % ? a%d -e& a e a%! 1ea- %2+ e1.I (a) &6e% (3#a) ,4 a a/&01 0 (3).

PROOF -e& (a) O% d,7,d,%* (3) ! @3#a) -e& *(3) e &6e 20&,e%&.A-40 ! 1e+a,%de1 &6e01e+ 56e% (3) ,4d,7,ded ! (3#a) &6e% &6e 1e+a,%de1 ,4 (a).&6e1e 01e (3) (3#a) *(3) < (a) (3) (3#a) *(3) &6e1e 01e (a) (*,7e% (3#a) ,4 aa/&01 0 (3).


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ALGEBRAIC IDENTITIES :

A-*e 1a,/ Ide%&,&,e4 S0+e /0++0%,de%&,&,e4 24ed &0 a/&01, e $0-!%0+,a-4(3


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ALGEBRAIC IDENTITIES :

A-*e 1a,/ Ide%&,&,e4 Ad7a%/ed ,de%&,&,e424ed &0 a/&01, e $0-!%0+,a-4 "# (3

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