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प्रश्न
Using the Remainder Theorem find the remainders obtained when ` x^3 + (kx + 8 ) x + k ` is divided by x + 1 and x - 2 .
Hence find k if the sum of the two remainders is 1.
उत्तर
Remainder theorem :
Dividend = Divisors × Quotient + Remainder
∴ Let f ( x ) = `x^3 + (kx + 8 ) x + k`
` = x^3 + kx ^2 + 8x + k`
Dividing f (x ) by x + 1 gives remainder as R1
∴ f (-1) = R1
Also , f ( 2 ) = R2
∴ f (-1) = (-1)3 + k (-1)2 + 8 (-1) + k
= -1 + k - 8 + k
=`2k - 9 = R_1`
`f (2) = (2)^3 + k (2)^2 + 8 xx 2 + k`
` = 8 + 4k + 16 + k`
` = 5 k + 24 = R_2`
Also,Sum of remainders = `R_1 + R_1 = 1`
∴ ( 2k - 9 ) + ( 5k +24 ) =1
7k + 15 = 1
7k = -14
k = - 2
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