Question

Seja P(x) = x^4 - (2a - 1)x^3 + (b -3)x² - 3x um polinômio de raízes 2 e -1. Determine a e b.

Answer 1

Olá, Adaordf.

$P(2) = 0\Rightarrow 2^4 - (2a - 1)\cdot2^3 + (b -3)\cdot2\² - 3\cdot2=\\\\=16-8(2a-1)+4(b-3)-6=10-16a+8+4b-12=\\\\=-16a+4b+6=0\Rightarrow8a-2b-3=0\Rightarrow8a-2b=3\\\\ P(-1) = 0\Rightarrow(-1)^4 - (2a - 1)\cdot(-1)^3 + (b -3)\cdot(-1)\² - 3\cdot(-1)=\\\\=1+2a-1+b-3+3=2a+b=0\Rightarrow4a+2b=0\\\\ \text{Sistema linear: }\begin{cases}8a-2b=3\\4a+2b=0\end{cases}(+)\Rightarrow12a=3\Rightarrow \boxed{a=\frac14}\Rightarrow \boxed{b=-\frac12}$