what must be added to polynomial 3x^4+5x^3-7x^2+5x+3 so that it is completely divisible by x^2+3x+1

Question

question from ch-2 polynomials class10th

Answer 1

what must be added to polynomial 3x^4+5x^3-7x^2+5x+3 so that it is completely divisible by x^2+3x+1 ?

Let p1 = 3x^4+5x^3-7x^2+5x+3

and p2 = x^2+3x+1

Then p1 divided by p2 = 3x^2 - 4x + 2 plus the remainder 3x + 1

for p1 to be completely divisible by p2, we need to subtract (3x + 1) from p1.

or, we should add (-3x - 1) to p1