Question

Решите уравнение: в)x^3+9x^2+23x+15=0

Answer 1

$x^{3} +9 x^{2}+23x+15=0 \\ x^{3} +5 x^{2} +4 x^{2} +20x+3x+15=0 \\ x^{2} (x+5)+4x(x+5)+3(x+5)=0 \\ ( x^{2} +4x+3)(x+5)=0 \\ x+5=0 \\ x_{1} =-5 \\ x^{2} +4x+3=0 \\ x_{2} = \frac{-4- \sqrt{16-12} }{2} = \frac{-4-2}{2}=-3 \\ x_{3} = \frac{-4+ \sqrt{16-12} }{2} = \frac{-4+2}{2} =-1 \\$
ответ х=-5;-3;-1

Answer 2

$x^3 + 9x^2 + 23x + 15 = 0$

Т.к. -1 + 9 - 23 + 15 = 0, корнем уравнения является $x= -1$

$x^3 + a_1x^2 + a_2x + a_3 = (x + 1)(x^2 + b_1x + b_2)\\\\ a_3 = b_2, b_2 = a_3 = 15\\\\ a_2 = b_2 + b_1, \ a_2 = a_3 + b_2, \ b_2 = a_2 - a_3 = 23 - 15 = 8\\\\ x^3 + 9x^2 + 23x + 15 = (x + 1)(x^2 + 8x +15)\\\\ x^2 + 8x + 15 = 0\\\\ x_1 + x_2 = -8 = -3 - 5\\\\ x_1*x_2 = 15 = (-3)*(-5)\\\\ x_1 = -3, x_2 = -5\\\\ \boxed{x_1 = -1, x_2= -3, x_3 = -5}$