Question

rozłóż wielomian na czynniki (skorzystaj ze wzorów skróconego mnożenia)

a)x^7-100x^5

b)49x^4-1/81

c)64x^10+x^2

d)x^2-6x+9

e)1/9x^2+1/3x+1/4

f)x^4-2x^2+1

g)(x+3)^2+2(x+3)+1

h) (x+1)^2-4

i)(x-3)^2-x^2

j)(x^2-6)^3-8

Answer 1

a)

x⁷ - 100x⁵ = x⁵(x² - 100) = x⁵(x² - 10²) = x⁵(x - 10)(x + 10)

 

b)

 

c)

64x¹⁰+x² = x²(64x⁸ + 1) = x²(4·16x⁸ + 1) = x²[4(2x²)⁴ + 1⁴] = x²(8x⁴ + 4x² + 1)(8x⁴ - 4x² + 1)

 

d)

x² - 6x + 9 = x² - 2·3x + 3² = (x - 3)² = (x - 3)(x - 3)

 

e)

f)

x⁴ - 2x² + 1 = (x²)² - 2 · x² + 1² = (x² - 1)² = (x² - 1)(x² - 1) = (x - 1)(x + 1)(x - 1)(x + 1)

 

g)

(x + 3)² + 2(x + 3) + 1 = (x + 3)² + 2·(x + 3)· 1 + 1² = [(x + 3) + 1]² = (x + 3 + 1)² = (x + 4)² = (x + 4)(x + 4)

 

h)

(x + 1)² - 4 = (x + 1)² - 2² = [(x + 1) - 2][(x + 1) + 2] = (x + 1 - 2)(x + 1 + 2) = (x - 1)(x + 3)

 

i)

(x - 3)² - x² = [(x - 3) - x][(x - 3) + x] = (x - 3 - x)(x - 3 + x) = -3(2x - 3)

 

j)

(x² - 6)³ - 8 = (x² - 6)³ - 2³ = [(x² - 6) - 2][(x² - 6)² + (x² - 6)·2 + 4] = (x² - 6 - 2)(x⁴ - 12x² + 36 + 2x² - 12 + 4) = (x² - 8)(x⁴ - 10x² + 28) = (x - 2√2)(x + 2√2)(x⁴ - 10x² + 28)