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ICSE Class 10 Answered

Consider the following polynominal, 
f (x)=x^3/sqrt (3)+x^2/sqrt (2)+x/sqrt (6)+2
 
g (x)=7x^4/6+5x^3/4+3x^2/2+1
 
When,f (x) is divided by x+sqrt (2),we get remainder R1.
 
Also when,g (x) is divided by x+1/2,we get remainder R2.
 Then , find the value of (R1+R2-R1R2).
Asked by nitinkumarsingh.singh.75 | 12 Aug, 2015, 10:40: AM
answered-by-expert 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 s f open parentheses x close parentheses equals fraction numerator x cubed over denominator square root of 3 end fraction plus fraction numerator x squared over denominator square root of 2 end fraction plus fraction numerator x over denominator square root of 6 end fraction plus 2 a n d g open parentheses x close parentheses equals fraction numerator 7 x to the power of 4 over denominator 6 end fraction plus fraction numerator 5 x cubed over denominator 4 end fraction plus fraction numerator 3 x squared over denominator 2 end fraction plus 1 G i v e n space t h a t space w h e n space f open parentheses x close parentheses space i s space d i v i d e d space b y space x plus square root of 2 comma space w e space g e t space r e m a i n d e r space R subscript 1 a n d w h e n space g open parentheses x close parentheses space i s space d i v i d e d space b y space x plus 1 half comma space w e space g e t space r e m a i n d e r space R subscript 2 T h u s comma space b y space r e m a i n d e r space t h e o r e m comma space w e space h a v e comma R subscript 1 equals f open parentheses negative square root of 2 close parentheses equals fraction numerator open parentheses negative square root of 2 close parentheses cubed over denominator square root of 3 end fraction plus fraction numerator open parentheses negative square root of 2 close parentheses squared over denominator square root of 2 end fraction plus fraction numerator open parentheses negative square root of 2 close parentheses over denominator square root of 6 end fraction plus 2 equals fraction numerator open parentheses negative 2 square root of 2 close parentheses over denominator square root of 3 end fraction plus fraction numerator 2 over denominator square root of 2 end fraction minus fraction numerator square root of 2 over denominator square root of 6 end fraction plus 2 equals fraction numerator open parentheses negative 2 square root of 2 close parentheses square root of 2 plus 2 square root of 3 minus square root of 2 plus 2 square root of 6 over denominator square root of 6 end fraction equals fraction numerator negative 4 plus 2 square root of 3 minus square root of 2 plus 2 square root of 6 over denominator square root of 6 end fraction S i m i l a r l y comma space w e space h a v e comma R subscript 2 equals g open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses equals fraction numerator 7 open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses to the power of 4 over denominator 6 end fraction plus fraction numerator 5 open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses cubed over denominator 4 end fraction plus fraction numerator 3 open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses squared over denominator 2 end fraction plus 1 equals fraction numerator 7 open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses to the power of 4 over denominator 6 end fraction plus fraction numerator 5 open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses cubed over denominator 4 end fraction plus fraction numerator 3 open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses squared over denominator 2 end fraction plus 1 equals 7 over 96 minus 5 over 32 plus 3 over 8 plus 1 equals fraction numerator 7 minus 15 plus 36 plus 96 over denominator 96 end fraction equals 124 over 96 equals 31 over 24 W e space n e e d space t o space f i n d space t h e space v a l u e space o f space R subscript 1 plus R subscript 2 minus R subscript 1 R subscript 2 colon R subscript 1 plus R subscript 2 minus R subscript 1 R subscript 2 equals fraction numerator negative 4 plus 2 square root of 3 minus square root of 2 plus 2 square root of 6 over denominator square root of 6 end fraction plus 31 over 24 minus open parentheses fraction numerator negative 4 plus 2 square root of 3 minus square root of 2 plus 2 square root of 6 over denominator square root of 6 end fraction close parentheses open parentheses 31 over 24 close parentheses rightwards double arrow R subscript 1 plus R subscript 2 minus R subscript 1 R subscript 2 equals fraction numerator 7 over denominator square root of 6 end fraction minus fraction numerator 7 over denominator 12 square root of 2 end fraction plus fraction numerator 7 over denominator 24 square root of 3 end fraction
Answered by Vimala Ramamurthy | 13 Aug, 2015, 08:57: AM

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