Question

rozłożyć na czynniki a)x³-x²-x+1= b)x³+54x²-2x-10= c)x³-7x²-3x+21= d)x³-3x²+3x-1= e)12x³+4x²+9x+3=

Answer 1

a)   x³-x²-x-1 = x²(x-1)-(x-1) = (x-1)(x²-1) = (x-1)(x+1)(x-1)

 

b)   x²-2x-10 = x²(x+5)-2(x+5) = (x+5)(x²-2) = (x+5)(x+√2)(x-√2)

 

c)   x³-7x²-3x+21 = x²(x-7)-3(x-7) = (x-7)(x²-3) = (x-7)(x+√3)(x-√3)

 

d)   x³-3x²+3x-1 = x³-1³-3x²+3x = (x-1)(x²+x+1²)-3x(x-1) = (x-1)(x²+x+1-3x) = 

   = (x-1)(x²-2x+1) = (x-1)(x-1)² = (x-1)(x-1)(x-1)

Lub

x³-3x²+3x-1 = (x-1)³ = (x-1)(x-1)(x-1)

   

 

e)   12x³+4x²+9x+3 = 4x²(3x+1)+3(3x+1) = (3x+1)(4x²+3)

 

 

Skorzystano ze wzorów:

(a+b)(a-b) = a²-b²

(a³-b³) = (a-b)(a²+ab+b²)

(a-b)³ = a³-3a²b+3ab²-b³