Question

Równania wielomianowe. a) 2x5-8x3+16x2-64=0 b) 3x5-12x3-12x2+48=0 c) 5x5+x3-6=30x2 d) 5=3x+5x4-3x5

Answer 1

a)

2x⁵ - 8x³ + 16x² - 64 = 0/ ÷2

 

x⁵ - 4x³ + 8x² - 32 = 0

x³(x² - 4) + 8(x² - 4) = 0

(x² - 4)( x³ + 8) = 0

x² - 4 = 0

x² = 4

x = 2 ∨ x = -2

x³ + 8 = 0

x³ = -8

x = -2

b)

3x⁵ - 12x³ - 12x² + 48 = 0

3x³(x² - 4) - 12(x² - 4) = 0

(x² - 4) ( 3x³ - 12) = 0

x² - 4 = 0

x = 2 ∨ x = -2

3x³ - 12 = 0

3x³ = 12

x³ = 4

x = ³√4

 

c)

5x⁵ + x³ - 6 = 30x²

5x⁵ + x³ - 30x² - 6 = 0

x³(5x² + 1) - 6(5x² + 1) = 0

(5x² + 1) (x³-6) = 0

5x² + 1 = 0

5x² ≠ -1 <- brak rozwiązań

x³-6 = 0

x³ = 6

x = ³√6

 

d)

5 = 3x + 5x⁴ - 3x⁵

3x⁵ - 5x⁴ - 3x + 5 = 0

x⁴(3x - 5) -1(3x - 5) = 0

(3x - 5) (x⁴ - 1) = 0

3x - 5 = 0

3x = 5

x = 5/3

 

x⁴ - 1 = 0

x⁴ = 1

x = 1