ОДЗ: x≠ -2
x³-11x²+39x-45=0
Делители 45: (+/-)1; (+/-)3; (+/-)5; (+/-)9; (+/-)15; (+/-)45
При х=3 3³-11*3²+39*3-45=0
x=3 - корень уравнения.
Делим x³-11x²+39x-45 на (x-3):
_ x³ -11x²+39x-45 | x-3
x³ - 3x² x²-8x+15
-------------
_ -8x²+39x
-8x²+24x
------------------
_ 15x-45
15x-45
--------------
0
x³-11x²+39x-45=(x-3)(x²-8x+15)
x²-8x+15=0
D=64-60=4
x₁=(8-2)/2=3
x₂=(8+2)/2=5
x²-8x+15=(x-3)(x-5)
x³-11x²+39x-45=(x-3)(x-3)(x-5)=(x-3)²(x-5)
(x-3)²(x-5)(x+2)≥0
x=3 x=5 x=-2
+ - - +
-------- -2 ---------- 3 ----------- 5 -------------
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x∈(-∞; -2)U{3}U[5; +∞)