Question

Rozłóż wielomian na czynniki
x³-½x²+4x-2
-12x³+4x²+9x-3
9x³-9x²+4x-4
-6x⁴+8x³-3x²+4x
-4x⁴+5x³+12x-15
8x³+1
x⁹-64x⁶
4x²+4x+1
3x²-2
16x⁴-1
(3x+1)²-x²

Answer 1

1)
x³ -(1/2)x² + 4x - 2 = x²(x -0,5) + 4(x - 0,5) =(x²+4)(x - 0,5)
2)
-12 x³ + 4 x² + 9 x - 3 = -4x²( 3x -1) +3(3x -1) = (3x -1)(3 - 4x²) =
= (3x - 1)(√3 -2x)(√3 +2x)
3)
9x³ - 9x² +4x - 4 = 9x²(x -1) + 4(x -1) = (9x² +4)(x -1)
4)
-6x⁴ + 8 x³ - 3 x² + 4x = -3x²(2x² + 1) + 4x(2x² + 1) =
= (-3x² + 4x)(2x² +1) = -x(3x - 4)(2x² + 1) = x(4 - 3x)(2x² + 1)
5)
-4x⁴ +5x³ + 12x - 15 = -4x(x³ - 3) +5(x³ -3) =(5 - 4x)(x³ - 3) =
= (5 - 4x)(x- ∛3)(x² +∛3x +(∛3)²)
6)
8x³ + 1 = (2x)³ + 1³ = (2x +1)(4x² +2x +1)
7)
x⁹ - 64 x⁶ = x⁶( x³ - 64) = x⁶(x³ - 4³) =
= x⁶(x -4)(x² +4x +16)
8)
4x² -4x + 1 = (2x - 1)² = (2x -1)(2x -1)
9)
3x² -2 = (√3x)² - (√2)² = (√3x - √2)(√3x + √2)
10)
(3x +1)² - x² = (3x+1 -x)(3x +1 +x) = (2x +1)(4x +1)
11)
16x⁴ -1 = (4x²)² - 1 = (4x² -1)(4x² + 1) = (2x -1)(2x +1)(4x² + 1)

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Answer 2

A)x³ -½x² + 4x - 2 = x²(x -½) + 4(x - ½) =(x²+4)(x - ½)

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

c)9x³ - 9x² +4x - 4 = 9x²(x -1) + 4(x -1) = (9x² +4)(x -1)

d)-6x⁴ + 8 x³ - 3 x² + 4x = -3x²(2x² + 1) + 4x(2x² + 1) =
= (-3x² + 4x)(2x² +1)

e)-4x⁴ +5x³ + 12x - 15 = -4x(x³ - 3) +5(x³ -3) =(5 - 4x)(x³ - 3) =

f)8x³ + 1 = (2x)³ + 1³ = (2x +1)(4x² +2x +1)

g)x⁹ - 64 x⁶ = x⁶( x³ - 64)

h)4x² -4x + 1 = (2x - 1)² = (2x -1)(2x -1)

i)3x² -2 = (√3x)² - (√2)² = (√3x - √2)(√3x + √2)

j)(3x +1)² - x² = (3x+1 -x)(3x +1 +x)

k)16x⁴ -1 = (4x²)² - 1 = (4x² -1)(4x² + 1)