Question

Dividindo um polinômio P(x) por x3 – x + 12, encontra-se como quociente x2 – 3x e resto –x – 4. O valor de P(x) é (A) x6 – 3x3 + 11x2 – 33x. (B) x6 – 3x3 + 11x2 – 34x – 4. (C) x5 – 3x4 – x3 + 15x2 – 36x. (D) x5 – 3x4 – x3 + 15x2 – 37x – 4. (E) x5 – 3x4 – x3 + 15x2 – 35x + 4.

Answer 1

$B(x) = x^3-x+12\\Q(x) = x^2-3x\\R(x)=-x-4\\\\gr(Q) = gr(P) - gr(B)\\2 = gr(P)- 3\\gr(P)=5\\\\P(x)\equiv B(x) \cdot Q(x)+R(x)\\ax^5+bx^4+cx^3+dx^2+ex+f \equiv (x^3-x+12)(x^2-3x)-x-4\\ax^5+bx^4+cx^3+dx^2+ex+f \equiv x^5-3x^4-x^3+3x^2+12x^2-36x-x-4\\ax^5+bx^4+cx^3+dx^2+ex+f \equiv x^5-3x^4-x^3+15x^2-37x-4\\\\a=1\\b=-3\\c=-1\\d=15\\e=-37\\f=-4\\\\ \boxed{\boxed{P(x)=x^5-3x^4-x^3+15x^2-37x-4}}$