Question

Determine o valor m para que o polinomio f = 2X3 + (1 − 2m)X2 + (1 − m)X + 1 seja divisıvel

por (2X+1).​

Answer 1

se f é divisível por 2x+1, então a raíz de 2x+1 é raíz de f, ou seja :

$\displaystyle \sf 2x+1=0\\\\ x = \frac{-1}{2} \\\\\\ Da{\'i}} \ \ f\left(\frac{-1}{2}\right) = 0 : \\\\\\\ 2\cdot \left(\frac{-1}{2} \right)^3+(1-2m)\cdot \left( \frac{-1}{2} \right)^2 +(1-m)\cdot \left(\frac{-1}{2}\right) +1 = 0 \\\\\\ \frac{-2}{8}+\frac{(1-2m)}{4}-\frac{(m-1)}{2}+1 = 0 \ \ \cdot( 8) \\\\\\ \frac{ 8\cdot (-2)}{8}+\frac{8\cdot (1-2m)}{4}-\frac{8\cdot (m-1)}{2}+8 = 0$

$\displaystyle \sf -2+2\cdot(1-2m)-4\cdot(m-1)+8=0 \\\\\ -2+2-4m-4m+4+8=0 \\\\ -8m + 12 = 0 \\\\ 8m =12 \frac{}{}\\\\ m=\frac{12}{8} \\\\\\ \boxed{\ \sf m = \frac{3}{2}\ }\checkmark$