1) When a polynomial is divided by -3x^5 + 10x - 11, what are the possible degrees of the remainder?
2) When a polynomial p(x) is divided by x + 1, the remainder is 5. When p(x) is divided by x + 5, the remainder is -7. Find the remainder when p(x) is divided by (x + 1)(x + 5).
+6248 1)
\(\text{The remainder can be any degree $0-4$}\)
2)
\(\text{by the polynomial remainder theorem}\\ p(-1)=5\\ p(-5)=-7\\~\\ p(x)\pmod{(x+1)(x+5)} = ax+b\\ p(x) = q(x)(x+1)(x+5) + (ax+b)\\ p(-1) = 5 = 0+(-a+b)\\ p(-5) = -7 = 0 + (-5a+b)\\ 4a= 12\\ a=3\\ b=8\\~\\ p(x) \pmod{(x+1)(x+5)} = 3x+8\)
