Question

Given the expression 2x^3 + mx^2 + nx + c leaves the same remainder when divided by x - 2 or x + 1, prove that m + n = -6

Answer 1

Given : Expression 2x³ + mx² + nx + c leaves the same remainder when divided by x - 2 or x + 1

To Find : prove that m + n = -6

Solution:

Polynomial p(x) divided by (x - a)

Then remainder = p(a)

P(x) =  2x³ + mx² + nx + c

divided by x - 2 or x + 1 leaves the same remainder

=> p(2)  = p(-1)

P(2) = 2(2)³ + m(2)² + n(2)+ c  = 16 + 4m + 2n + c

P(-1) = 2(-1)³ + m(-1)² + n(-1)+ c  = -2 + m  - n  + c

16 + 4m + 2n + c  =  -2 + m  - n  + c

=> 3m + 3n = -18

=> m + n = - 6

QED

Hence Proved

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