Question

Rozwiąż równania wielomianowe:

a) x³-10x+9=0

b) 3x⁴-10x³+10x-3=0

Answer 1

a)

x³-10x+9=0

x³ - x - 9x + 9 = 0

x(x² - 1) - 9(x - 1) = 0

x(x - 1)(x + 1) - 9(x - 1) = 0

(x - 1) [ x(x  +1) - 9 ] = 0

(x - 1) [ x²  + x - 9 ] = 0

x - 1 = 0   lub   x² + x - 9 = 0

x = 1               Δ = 1 + 36 = 37

                        √Δ = √37

                         x1 = (-1 -  √37) / 2

                         x2 = (-1 +  √37) / 2

 

b)

3x⁴ - 10x³ + 10x - 3 = 0

3x⁴ - 3 - 10x³ + 10x = 0

3(x⁴ - 1) - 10x(x² - 1) = 0

3(x² - 1)(x² + 1)  - 10x(x² - 1) = 0

(x² - 1) [ 3(x² + 1) - 10x ] = 0

x² - 1 = 0    lub  3x² + 3 - 10x = 0

x² = 1                    3x² - 10x + 3 = 0

x = 1  lub  x= -1    Δ = 100 - 36 = 64

                               √Δ = 8

                                  x1 = (10 - 8)/6 = 2/6 = 1/3

                                  x2 = (10 + 8)/6 = 18/6 = 3