Question

Rozłóż wielomian na czynniki:
a) x³-2x²+2x-4
b) 3x³+6x²-x-2
c)9x3-9x²+4x-4
d)-6x⁴+8x³-3x²+4x

Rozłóz wielomian na czynniki jak najniższych stopni
a)-3x²+5x+2
b)-4x³-4x²+3x
c) (2x+3)(x+1)-(x-1)(3x-2)
d)(x²+x-6)(x²+x+5)
e)(9x²-12x+4)(3x²-4x+4)

Answer 1

A) x³-2x²+2x-4 = x²(x-2) + 2(x-2) = (x-2)(x²+2)
b) 3x³+6x²-x-2 = -3x²(-x-2) + (-x-2) = (-x-2)(-3x² + 1)
c)9x³-9x²+4x-4 = 9x²(x - 1) + 4(x-1) = (x-1)(9x²+4)
d)-6x⁴+8x³-3x²+4x = 2x³(-3x + 4) + x(-3x + 4) = (-3x+4)(2x³ + x) = (-3x+4)*x(2x²+1) = x(-3x+4)(2x²+1)


a)-3x²+5x+2
Δ=5² - 4 * (-3)*2 = 25 +24 = 49
√Δ = √49 = 7
x₁=-5-7/2*(-3) = -12/-6 =2
x₂=-5+7/2*(-3) = 2/-6 = -1/3

-3x²+5x+2 = -3(x-2)(x+⅓)

b)-4x³-4x²+3x = x(-4x²-4x + 3)
Δ = (-4)² - 4 * (-4) *3 = 16 + 48 = 64
√Δ = √64 = 8
x₁= 4 - 8/2*(-4) = -4/-8 = 1/2 = 0,5
x₂= 4 + 8 / 2 *(-4) = 12/-8 = -1,5

-4x³-4x²+3x = x(-4x²-4x + 3) = x * (-4)(x-0,5)(x+1,5) = -4x(x-0,5)(x+1,5)


c) (2x+3)(x+1)-(x-1)(3x-2) = 2x² + 2x + 3x + 3 - (3x² - 2x - 3x+2) = 2x² + 5x + 3 - (3x²-5x + 2) = 2x²+5x+3 - 3x²+5x - 2 = -x²+10x + 1
Δ = 10² - 4* (-1) * 1 = 100 +4 = 104
√Δ = √104 = √4 * 26 = 2√26
x₁= -10 - 2√26 / 2 * (-1) = -2(5+√26) / -2 = 5+√26
x₂= -10 + 2√26 / 2 * (-1) = -2(5-√26) / -2 = 5 - √26

-x²+10x + 1 = -(x -(5+√26))(x-(5-√26)) = -(x-5-√26)(x-5+√26)

d)(x²+x-6)(x²+x+5)
Δ₁= 1²-4*1*(-6) = 1 + 24 = 25
√Δ₁= √25 = 5
x₁= -1 - 5 / 2*1 = -6/2 = -3
x₂= -1 + 5 / 2*1 = 4/2 = 2

x²+x-6 = (x+3)(x-2)

Δ₂= 1² - 4 * 1 * 5 = 1 - 20 = -19 brak pierwiastków

(x²+x-6)(x²+x+5) = (x+3)(x-2)(x²+x+5)

e)(9x²-12x+4)(3x²-4x+4)
Δ₁= (-12)² - 4 * 9 * 4 = 144 - 144 = 0
x = 12/2*9 = 12/18 = 2/3
9x²-12x+4 = 9(x+⅔)²

Δ₂= (-4)² - 4 * 3 * 4 = 16 - 48 = -32 brak pierwiastków

(9x²-12x+4)(3x²-4x+4) = 9(x+⅔)²(3x²-4x+4)