Question

Let p(x) be a polynomial. When it is divided by (x-19) and (x-91), then the remainders are 91 and 19 respectively The remainder when p(x) is divided by (x-19)(x-91) is- (a) x - 110 (b) 110 (c) 110 - x (d) 4x + 88

Answer 1

the formula is , for a polynomial f(x) when divided by (x-a) and (x-b)
remainder is , r(x) = ((f(a) - f(b)/a-b)x + ( bf(a) - af(b) )/a-b

so the remainder in this case is r(x) = ((91 - 19)/19 - 91)x + (91 x 91 - 19 x 19)/91 - 19 = -x + 110 = 110 - x

Answer 2

let, the polynomial be f(x) × (x - 19)(x - 91) + (ax + b)
where, ax+b is the remainder.
Now, acc. to remainder theorem,
p(19) = 91
or, 19a + b = 91 _____(1)
also, p(91) = 19
or, 91a+b = 19_______(2)
subtracting 1 from 2, we get,
72a = -72
a = -1
again, -19 + b = 91
b = 110
so, the remainder is
ax+ b
= -x + 110
= 110 - x