Question

The polynomial ax2 + bx + c is divisible by 2x – 1 and x + 2. When the polynomial is divided
by x-2, the remainder is 12. Find a = ?​

Answer 1

f(x) is divisible by g(x), then the ‘remainder’ of a long division would equal zero.

The remainder after dividing f(x) by g(x) is

[(ab² – ac + b)x + c(ab – 1)]/(x² + bx + c)

For the numerator to equal zero, both terms in parentheses have to be equal to zero

ab² – ac + b = 0 and ab – 1 = 0

ab = 1…….(i)

or a=1/b

ab² – ac + b = 0

ac = ab² + b

ab = 1………..(from (i))

ac = b + b = 2b

c=2b/a

c = 2b/(1/b) substituting value of a

or c = 2b2