What must be added to the polynomial p(x) = 5x4 + 6x3 - 13x3 - 44x + 7 so that the resulting polynomial is exactly divisible by the polynomial q(x) = x2 + 4x + 3 and the degree of the polynomial to be added must be less than degree of the polynomial q(x).
We know that,
Dividend = Divisor × Quotient + Remainder
Here, Dividend = p (x) = 5x4 + 6x3 – 13x2 – 44x + 7
Divisor = q (x) = x2 + 4x + 3
So,
Remainder = –114x –77,
So, – (–114x – 77) = 114x + 77 Should be added to p (x) to make it exactly divisible by q (x).