Question

Zadanie na jutro! prosze o pomoc :* Rozłóż wielomiany na czynniki , korzystając z postaci iloczynowej funkcji kwadratowej: a)x2-4x+3 b)x2-6x+5 c)9x2-30x+25 d)6x2+7x-3 e)x4+x2-20 f) x4-10x2+9

Answer 1

postać  iloczynowa funkcji kwadratowej :

∆ > 0

y = a(x-x₁)(x-x₂)

∆ = 0

y = a (x-x₁)²

∆< 0

nie istnieje

 

a)

 x² - 4x  +3  = (x-1)(x-3)

 

a=1,  b =- 4  , c= 3

 Δ = b²-4ac =(- 4)² - 4*1*3  =16 - 12 = 4

√∆ = √4 = 2

x₁ = (-b-√∆)2a  = (4 -2)/2 = 2/2 = 1

x₂ = (-b +√∆)/2a = (4 + 2)/2 = 6/2 = 3

y = (x-1)(x-3)

 

b) x² - 6x + 5 =  (x-1)(x-5)

 

a=1,  b =- 6  , c= 5

 Δ = b²-4ac =(- 6)² - 4*1*5  =36 - 20 = 16

√∆ = √16 = 4

x₁ = (-b-√∆)2a  = (6 -4)/2 = 2/2 = 1

x₂ = (-b +√∆)/2a = (6 + 4)/2 = 10/2 = 5

y = (x-1)(x-5)

 

 

c)

9x²- 30x + 25 =  9(x+5/3)²

 

a=9,  b =- 30  , c= 25

 Δ = b²-4ac =(- 30)² - 4*9*25  = 900 – 900 = 0

x₁ = -b/2a  =  30/18 = -5/3

y = 9(x+5/3)²

 

d)

6x² + 7x – 3 =6(x+3/2)(x- 1/3)

 

a=6,  b =7  , c= -3

 Δ = b²-4ac = 7² - 4*6*(-3)  = 49 + 72 = 121

√∆ = √121 = 11

x₁ = (-b-√∆)2a  = (-7 -11)/12 = -18/12 = - 3/2

x₂ = (-b +√∆)/2a = (-7 +11)/12 = 4/12 = 1/3

y = 6(x+3/2)(x- 1/3)

 

e)

x⁴ + x² -20 = (x² +5)(x² -4) = (x² -5)(x-2)(x+2)

 

x² = t

t² + t -20

Δ = b²-4ac = 1 – 4 * 1 * (-20)   = 1 + 80 = 81

√∆ = √81= 9

x₁ = (-b-√∆)2a  = (-1-9)/2= -10/2  = - 5

x₂ = (-b +√∆)/2a =(-1+9)/2= 8/2  =  4

 

y = (x² +5)(x² -4) = (x² -5)(x-2)(x+2)

 

 

f)

x⁴ - 10x² + 9 =  (x-3)(x+3)(x-1)(x+1)

 

x² = t

t² - 10 t + 9

Δ = b²-4ac = (-10)² – 4 * 1 * 9   = 100 – 36 = 64

√∆ = √64= 8

x₁ = (-b-√∆)2a  = (10-8)/2=  2/2  = 1

x₂ = (-b +√∆)/2a =(10+8)/2=18/2  =  9

 

y = (x² -9)(x² -1) = (x-3)(x+3)(x-1)(x+1)