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ucenikthl
@ucenikthl
November 2021
1
16
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Разложите на множители
1)a^3-27+3a^2-9a
2)b^3+125+2b+10
3)3x+6y-x^3-8y^3
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Verified answer
1) a
³ - 27 + 3a² - 9a = (a³ - 3³) + 3a(a - 3) =
= (a - 3)(a
² + 3a + 3²
) + 3a(a - 3) = (a - 3)(a
² + 3a + 9 + 3a
) =
= (a - 3)(a
² + 6a + 9
) = (a - 3)(a + 3)
²
2) b
³ + 125 + 2b + 10 = b³ + 5³ + 2(b + 5) =
= (b + 5)(b
² - 5b + 5²
) + 2(b + 5) = (b + 5)(b
² - 5b + 25 + 2
) =
= (b + 5)(b
² - 5b + 27
)
3) 3x + 6y - x
³ - 8y³ = 3(x + 2y) - (x³ + (2y)³) =
= 3(x + 2y) - (x + 2y)(x² - 2xy + (2y)²) = (x + 2y)(3 - (x² - 2xy + 4y²)) =
= (x + 2y)(3 - x² + 2xy - 4y²)
---------------------------------------------
Использованы формулы суммы и разности кубов
c³ + d³ = (c + d)(c² - cd + d²)
c³ - d³ = (c - d)(c² + cd - d²)
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Answers & Comments
Verified answer
1) a³ - 27 + 3a² - 9a = (a³ - 3³) + 3a(a - 3) == (a - 3)(a² + 3a + 3²) + 3a(a - 3) = (a - 3)(a² + 3a + 9 + 3a) =
= (a - 3)(a² + 6a + 9) = (a - 3)(a + 3)²
2) b³ + 125 + 2b + 10 = b³ + 5³ + 2(b + 5) =
= (b + 5)(b² - 5b + 5²) + 2(b + 5) = (b + 5)(b² - 5b + 25 + 2) =
= (b + 5)(b² - 5b + 27)
3) 3x + 6y - x³ - 8y³ = 3(x + 2y) - (x³ + (2y)³) =
= 3(x + 2y) - (x + 2y)(x² - 2xy + (2y)²) = (x + 2y)(3 - (x² - 2xy + 4y²)) =
= (x + 2y)(3 - x² + 2xy - 4y²)
---------------------------------------------
Использованы формулы суммы и разности кубов
c³ + d³ = (c + d)(c² - cd + d²)
c³ - d³ = (c - d)(c² + cd - d²)