Представить в виде произведения: m³-m²n-mn²+n³ x⁴+x³-x-1 a⁴+a³+a+1 (

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а) z* = -z·i

z = x + iy

x - iy = -(x + iy)·i

x - iy = -ix + y

x + ix = y + iy

x·(1 + i) = y·(1 + i)

y = x

z = x + ix, x ∈ R

б) 2·|z| - 8z + 1 + 2i = 0

z = x + iy

2√(x² + y²) - 8·(x + iy) + 1 + 2i = 0

2√(x² + y²) - 8x - i8y + 1 + 2i = 0

2√(x² + y²) = (8x - 1) + i(8y - 2)

2√(x² + y²) = 8x - 1

8y - 2 = 0

y = 1/4

2√(x² + (1/4)²) = 8x - 1

4(x² + 1/16) = 64x² - 16x + 1

8x - 1 ≥ 1/2

4x² + 1/4 = 64x² - 16x + 1

8x ≥ 3/2

60x² - 16x + 3/4 = 0

x ≥ 3/16

240x² - 64x + 3 = 0

D = 64² - 4·240·3 = 1216

x = (64 (+/-) √1216)/480 = (64 (+/-) 8√19)/480 = (8 (+/-) √19)/60

x = 2/15 (+/-) √19/60

x ≥ 3/16

x = 2/15 + √19/60

z = 2/15 + √19/60 + i/4

а) z* = -z·i

z = x + iy

x - iy = -(x + iy)·i

x - iy = -ix + y

x + ix = y + iy

x·(1 + i) = y·(1 + i)

y = x

z = x + ix, x ∈ R

б) 2·|z| - 8z + 1 + 2i = 0

z = x + iy

2√(x² + y²) - 8·(x + iy) + 1 + 2i = 0

2√(x² + y²) - 8x - i8y + 1 + 2i = 0

2√(x² + y²) = (8x - 1) + i(8y - 2)

2√(x² + y²) = 8x - 1

8y - 2 = 0

y = 1/4

2√(x² + (1/4)²) = 8x - 1

4(x² + 1/16) = 64x² - 16x + 1

8x - 1 ≥ 1/2

4x² + 1/4 = 64x² - 16x + 1

8x ≥ 3/2

60x² - 16x + 3/4 = 0

x ≥ 3/16

240x² - 64x + 3 = 0

D = 64² - 4·240·3 = 1216

x = (64 (+/-) √1216)/480 = (64 (+/-) 8√19)/480 = (8 (+/-) √19)/60

x = 2/15 (+/-) √19/60

x ≥ 3/16

x = 2/15 + √19/60

z = 2/15 + √19/60 + i/4