Question

Se a soma de duas raízes de P(x)= x³-6x²+11+k é 3 ,então o número real k quanto vale ? ​

Answer 1

$\sf P(x)=x^3-6x^2+11+k \\\\ raizes : \\\ x_1,\ x_2,\ x_3 \\\\ Usando\ as\ rela{\c c}{\~o}es \ de\ girard : \\\\ x_1+x_2+x_3 = \frac{-(-6)}{1}\\\\ x_1+x_2+x_3 = 6 \\\\ portanto\ a\ soma\ de\ duas\ das \ raizes \ \'e \ 3,\ ou \ seja : \\\\ \underbrace{\sf x_1+x_2}_{3}+x_3=6 \\\\ 3+x_3=6 \\\\ x_3 = 3$

Daí :

P(3) = 0

$\sf 3^3-6.3^2+11+k = 0 \\\\ 27-6.9+11+k=0\\\\ 27-54+11+k=0\\\\ k = 54-27-11 \\\\ \boxed{\sf k = 16}\checkmark$