Question

Determinati radacinile polinomului f

$f=X^{3}-4X^{2} +3X+2$

Answer 1

2. Se consideră polinomul  f = X^3 - 4X^2 + mX + 2, unde m este număr real.

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c) Pentru m = 3 , determinați rădăcinile polinomului f.

 $\displaystyle\\x^3-4x^2+3x+2=0\\x^3-2x^2-2x^2+4x-x+2=0\\x^2(x-2)-2x(x-2)-(x-2)=0\\(x-2)(x^2-2x-1)=0\\\\x-2 = 0~~~\text{sau}~~~x^2-2x-1=0\\\\\boxed{\bf~x_1=2}\\\\x_{23}=\frac{2\pm\sqrt{4+4}}{2}=\frac{2\pm\sqrt{8}}{2}=\frac{2\pm2\sqrt{2}}{2}=1\pm\sqrt{2}\\\\\boxed{\bf~x_2=1+\sqrt{2}}\\\\\boxed{\bf~x_3=1-\sqrt{2}}$

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