f4數學題一問?10點

Let f(x) be the polynomial.

x⁶ - x²
= x²(x⁴ - 1)
= x²(x² - 1)(x² + 1)
= x²(x + 1)(x - 1)(x² + 1)

x² + x
= x(x + 1)

3x⁷ + 3x⁶ - 3x³ - 3x²
= 3x²(x⁵ + x⁴ - x - 1)
= 3x²[x⁴(x + 1) - (x + 1)]
= 3x²(x + 1)(x⁴ - 1)
= 3x²(x + 1)²(x - 1)(x² + 1)

The product of two polynomials = H.C.F. x L.C.M.
(x⁶ - x²) * f(x) = (x² + x) * (3x⁷ + 3x⁶ - 3x³ - 3x²)
x²(x + 1)(x - 1)(x² + 1) * f(x) = x(x + 1) * 3x²(x + 1)²(x - 1)(x² + 1)
x²(x + 1)(x - 1)(x² + 1) * f(x) = 3x³(x + 1)³(x - 1)(x² + 1)
Hence, f(x) = 3x(x + 1)² = 3x³ +6x² +3x

Hence, the polynomial is 3x³ +6x² +3x.