Polynomials

Consider the function g(x)=x4+2x3+px2+qx−4, where p,q∈R are constants. When g(x) is divided by (x−1), the remainder is −12, and when divided by (x+3) the remainder is 20. Find the values of p and q.

When x3−ax2+bx−2 is divided by x+3, the remainder is −8. When divided by x−2, the remainder is 22. Find the values of a and b.

Given that (x+3) is a factor of P(x) and that P(x) leaves a remainder of 6 when divided by (x+2), find the remainder when P(x) is divided by x2+5x+6.

The polynomial P(x)=3x4+ax3+bx2+25x−10, where a,b∈R, has a factor 3x2+5x−2. Find the values of a and b.

The cubic polynomial Ax3+Bx2−7x−12 is exactly divisible by x2+4x+3. Find the values of A and B.