Question

$P(x) = x³ -4x²+ax+b,unde ,a ,b ,x ∈ R$
a) detrminati a si b , stiind ca P (1) = -3 si P(-1)= 7
b) a=-6 si b=6 , descompuneti in produs de trei factori P(x) +9x-6.

Answer 1

P(x)=x³-4x²+ax+b
P(1)=1³-4·1²+a+b=1-4+a+b=a+b-3
a+b-3=-3⇒a+b=-3+3⇒a+b=0
P(-1)=(-1)³-4·(-1)²+a·(-1)+b=-1-4-a+b=-a+b-5
-a+b-5=7⇒  -a+b=7+5⇒ -a+b=12
avem sistemul
{a+b=0
{-a+b=12 
⇒a-a+b+b=12
2b=12
b=12/2
b=6
din a+b=0⇒a=-b deci a=-6
b) P(x)=x³-4x²-6x+6
P(x)+9x-6=x³-4x²-6x+6+9x-6=x³-4x²+3x= x(x²-4x+3)=x(x-1)(x-3)