Вправи: 938

938. Розкладіть на множники многочлен:

1) x³ + Зх² + 2х = х(x² + Зх + 2).

x² + Зх + 2 = 0; D = З² – 4 • 2 = 1;

х₁ = -3+12 = –1; x₂ = -3-12 = –2.

х(х² + Зх + 2) = х(х + 1)(х + 2).

2) –2х³ – 5х² + Зх = –х(2х² + 5х – 3).

2x² + 5х – 3 = 0; D = 5² – 4 • 2 • (–3) = 49;

х₁ = -5+74 = 12; x₂ = -5-74 = –3.

–x(2x² + 5x – 3) = –2x(x – 12)(x + 3) = –x(2x – 1)(x + 3) = x(x + 3)(1 – 2x).

3) 14x⁴ + x³ – 54x² = 14x²(x² + 4x – 5).

x² + 4x – 5 = 0; D = 16 + 4 • 5 = 36;

x₁ = -4+6 2 = 1; x₂ = -4-62 = –5.

14x²(x² + 4x – 5) = 14x²(x – 1)(x + 5).

4) –12x⁵ + 2x⁴ + 6x³ = –12x³(x² – 4x – 12).

x² – 4x – 12 = 0; D = (–4)² – 4 • (–12) = 64;

x₁ = 4+82 = 6; x₂ = 4-82 = –2.

–12x³(x² – 4x – 12) = –12x³(x – 6)(x + 2).