Question

what must be subtracted from(x^4+2x^3-4x+6) so that the result is exactly divisible by (x^2+2x-3)?​

Answer 1

Let,

→  p(x) = x⁴ + 2x³ - 4x + 6

→  d(x) = x² + 2x - 3

Let the polynomial which should be subtracted from  p(x)  [our answer]  be r(x).

Let the quotient polynomial which is obtained on dividing  p(x) - r(x)  by  d(x)  be  q(x).

Given that if we subtract  r(x)  from  p(x),  then the resultant polynomial divided by  d(x)  gives quotient  q(x).  This means  p(x)  gives quotient  q(x)  and remainder  r(x)  on division by  d(x). This can be expressed as,

p(x) = q(x)d(x) + r(x)

So, dividing  p(x)  by  d(x),

    p(x)

⇒  x⁴ + 2x³ - 4x + 6

⇒  x⁴ + 2x³ - 3x² + 3x² + 6x - 10x - 9 + 15

⇒  x⁴ + 2x³ - 3x² + 3x² + 6x - 9 - 10x + 15

⇒  x²(x² + 2x - 3) + 3(x² + 2x - 3) - 10x + 15

⇒  (x² + 2x - 3)(x² + 3) - 10x + 15

⇒  d(x)q(x) + r(x)

From this, we get,

→  q(x) = x² + 3

→  r(x) = - 10x + 15

Hence,  - 10x + 15  is the answer.