Question

Se P(x) =2x³-4x²+ax+b e Q(x)=2x²-x-1 são polinomios , os valores de a e b , para que p(x) seja divisivel por Q(x) são respectivamente: 

gabarito: 1/2 e 2/3

Answer 1

$.~~2x^3-4x^2+ax+b~|~2x^2-x-1$
$-2x^3+x^2+x~~~~~~~~~~~x-\dfrac{3}{2}$
$---------$
$.~~~~~~-3x^2+ax+x+b$
$.~~~~~~+3x^2-\dfrac{3x}{2}-\dfrac{3}{2}$
$---------------$
$.~~~~~~~~~~~~~~~~ax+x-\dfrac{3x}{2}+b-\dfrac{3}{2}$

Para que o resto seja $0$ devemos ter $ax+x-\dfrac{3x}{2}+b-\dfrac{3}{2}=0$

Logo:

$ax+x-\dfrac{3x}{2}=0 \iff ax+x=\dfrac{3x}{2} \iff a+1=\dfrac{3}{2}$
 
$2a+2=3 \iff 2a=1 \iff \boxed{a=\dfrac{1}{2}}$

$b-\dfrac{3}{2} \iff \boxed{b=\dfrac{3}{2}}$

Answer 2

P(x) =2x³-4x²+ax+b e Q(x)=2x²-x-1

Se P(x) é divisível por Q(x) ==>as raízes de Q(x) são raízes de P(x)

2x²-x-1 =0
x'=[1+√9]/4=1
x''=[1-√9]/4=-1/2

P(1)=2-4+a+b=0    =>a+b=2(i)
p(-1/2)=-1/4-1-a/2+b=0 ==>-a/2+b=5/4 (ii)

(i)-(ii)

3a/2=3/4  ==>a=1/2

Usando a+b=2
1/2+b=2
b=2-1/2=3/2