Q.65. Polynomial equation, Remainder theorem

Question 65.
When P(x) is divided by x^2-3x+7, the quotient is x^2-3x+7 and the remainder is unknown. However, when P(x) is divided by x - 2 the remainder is 29 and when divided by x + 1, the remainder is -16. If the remainder has the form ax+b, find a and b.

Answer 65.
P(x) / (x^2 - 3x + 7) = x^2 - 3x + 7 + (ax + b)/(x^2 - 3x + 7)
=> P(x)
= (x^2 - 3x + 7)^2 + ax + b
= x^4 - 6x^3 + 23x^2 - 42x + 49 + ax + b

When divided by x-2, the remainder is 29
=> P(2) = 29
=> 2^4 - 6*2^3 + 23*2^2 - 42*2 + 49 + 2a + b = 29
=> 16 - 48 + 92 - 84 + 49 + 2a + b = 29
=> 2a + b = 4 ... ( 1 )

When divided by x + 1, the remainder is - 16
=> P(-1) = - 16
=> (-1)^4 - 6*(-1)^3 + 23*(-1)^2 - 42*(-1) + 49 + a*(-1) + b = -16
=> 1 + 6 + 23 + 42 + 49 - a + b = -16
=> a - b = 137 ... ( 2 )

Solving equtions ( 1 ) and ( 2 ),
a = 47 and b = - 90.

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