Question

Rozwiąż nierówność

Answer 1

3x³+2x²-7x+2 > 0
3+2-7+2 = 0 => x₁ = 1
3x³ + 2x² - 7x+ 2 =
=  3x³ + 2x² - 2x-5x + 2 = 3x³ + 2x(x-1) - 5x + 2 =
= 3x³ + 2x(x-1) -3x-2x + 2 =
= 3x(x²-1)+ 2x(x-1)-2(x-1) = 3x(x-1)(x+1)+ 2x(x-1)-2(x-1) = (x-1)[3x(x+1)+2x-2] =
= (x-1)(3x²+5x-2)
3x²+5x-2
D = 25+24=49
x₂ = (-5-7)/6 = -2
x₃ = (-5+7)/6 = 1/3
3x³+2x²-7x+2 = (x-1)(3x²+5x-2) = (x-1)(x+2)(x-1/3)

y = (x-1)(x+2)(x-1/3) > 0
dla x∈(-∞; -2)
(-)(-)(-)=(-)   => y < 0 nie!
dla x∈(-2; 1/3)
(-)(+)(-) = (+) => y > 0 ok!
dla x∈(1/3; 1)
(-)(+)(+)=(-)   => y < 0 nie!
dla x∈(1; +∞)
(+)(+)(+)=(+)   => y > 0 ok!

x ∈ (-2; 1/3)∪(1; +∞)