Question

Se -1 é raíz do polinomio P(x) = x³ - 4x² + x - k , k ∈ R, então as duas outras raízes são:

a) reais e de multiplicidade 2.
b) racionais e negativas.
c) não reais.
d) irracionais.
e) inteiras.

Answer 1

$P(x)=x^3-4x^2+x-k\\\\P(-1)=0\\\\P(-1)=(-1)^3-4\cdot(-1)^2+(-1)-k\\\\-1-4\cdot1-1-k=0\\-1-4-1-k=0\\-k=6\\\,\,k=-6\\\\P(x)=x^3-4x^2+x-(-6)\\P(x)=x^3-4x^2+x+6\\\\P(x)=(x+1)\cdot{q_1(x)}\\\\(x+1)\cdot{q_1}(x)=0\\\\Determinando\,\,q_1(x)\,\,temos:\\\\q_1(x)=x^2-5x+6\\\\q_1(x)=0\\\\x^2-5x+6=0\\\\a=1\,\,\,\,\,\,\,\,\,\,\,b=-5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,c=6\\\\\triangle=b^2-4ac\\\triangle=(-5)^2-4\cdot{1}\cdot{6}\\\triangle=25-24\\\triangle=1\\\\x= \dfrac{-b \frac{+}{-} \sqrt{\triangle} }{2a}$

$x=\dfrac{-(-5) \frac{+}{-}1 }{2}\\\\\\x'= \dfrac{5+1}{2}\\\\\\x'=3 \\\\\\x"= \dfrac{5-1}{2}\\\\ \\x"=2\\\\\\q_1(x)=(x-3)\cdot(x-2)$

$\boxed{S=\{-1,2,3\}}$

Veja que as outras duas raízes são números inteiros (2,3)

LETRA E