Cho đa thức M(x) = 3x⁵ – 4x³ + 9x + 2. Tìm các đa thức N(x), Q(x) sao cho: N(x) – M(x) = –5x⁴ – 4x³ + 2x² + 8x và Q(x) + M(x) = 3x⁴ – 2x³ + 9x² – 7.

• Ta có: N(x) – M(x) = –5x⁴ – 4x³ + 2x² + 8x

Suy ra N(x) = –5x⁴ – 4x³ + 2x² + 8x + M(x)

Do đó N(x) = –5x⁴ – 4x³ + 2x² + 8x + (3x⁵ – 4x³ + 9x + 2)

                    = –5x⁴ – 4x³ + 2x² + 8x + 3x⁵ – 4x³ + 9x + 2

                    = 3x⁵ – 5x⁴ + (–4x³ – 4x³) + 2x² + (8x + 9x) + 2

                    = 3x⁵ – 5x⁴ – 8x³ + 2x² + 17x + 2.

• Ta có: Q(x) + M(x) = 3x⁴ – 2x³ + 9x² – 7.

Suy ra Q(x) = 3x⁴ – 2x³ + 9x² – 7 – M(x)

Do đó Q(x) = 3x⁴ – 2x³ + 9x² – 7 – (3x⁵ – 4x³ + 9x + 2)

                    = 3x⁴ – 2x³ + 9x² – 7 – 3x⁵ + 4x³ – 9x – 2

                    = – 3x⁵ + 3x⁴ + (– 2x³ + 4x³) + 9x² – 9x + (– 7 – 2)

                    = – 3x⁵ + 3x⁴ + 2x³ + 9x² – 9x – 9.

Vậy N(x) = 3x⁵ – 5x⁴ – 8x³ + 2x² + 17x + 2;

Q(x) = – 3x⁵ + 3x⁴ + 2x³ + 9x² – 9x – 9.