Разложите на множители трехчлен: а) х⁴ – 47х² – 98; б) х⁴ – 85х² + 1764.

Решение:

а) х⁴ – 47х² – 98

Замена x² = b

b² = 47b – 98

D = (-47)² – 4 ⋅ 1 ⋅ (-98) = 2209 + 392 = 2601 больше 0

√D = √2601 = √51² = 51

b₁ = -(-47) – 51 / 2 = 47 – 51 / 2 = -4 / 2 = -2

b₂ = -(-47) + 51 / 2 = 47 + 51 / 2 = 98 / 2 = 49

b² = – 47b – 98 = (b – (-b)) ⋅ (b – 49) = (b + 2) ⋅ (b – 49)

x⁴ – 47x² – 98 = (x² + 2) ⋅ (x² – 49) = (x² + 2) ⋅ (x – 7) ⋅ (x + 7)

б) х⁴ – 85х² + 1764

Замена x² = b

b² – 85b + 1764

D = (-85)² – 4 ⋅ 1 ⋅ 1764 = 7225 – 7056 = 169 больше 0

√D = √169 = √13² = 13

b₁ = -(-85) – 13 / 2 = 85 – 13 / 2 = 72 / 2 = 36

b₂ = -(-85) + 13 / 2 = 85 + 13 / 2 = 98 / 2 = 49

b² – 85b + 1764 = (b – 36) ⋅ ( b – 49)

x⁴ – 85x² + 1764 = (x² – 36) ⋅ ( x² – 49) = (x – 6) ⋅ (x + 6) ⋅ (x – 7) ⋅ (x + 7)

Источник : Учебник по Алгебре 9 класса авторов Ю.Н. Макарычев, Н.Г. Миндюк, К.И. Нешков, С.Б. Суворова, 2014г.