Đáp án:
`A = x(x + 1)(x - 1)(x + 2) - 3`
`= [x(x + 1)][(x - 1)(x + 2)] -3`
`= (x^2 + x)(x^2 + 2x - x - 2) - 3`
`= (x^2 + x)(x^2 + x - 2) - 3`
Đặt `x^2 + x = a`, ta được:
`A = a(a - 2) - 3`
`= a^2 - 2a - 3`
`= a^2 + a - 3a - 3 = a(a + 1) - 3(a + 1)`
`= (a + 1)(a - 3)`
`= (x^2 + x + 1)(x^2 + x - 3)`
`B = (x - 7)(x - 5)(x - 4)(x - 2) - 72`
`= [(x - 7)(x - 2)][(x - 5)(x - 4)] - 72`
`= (x^2 - 2x - 7x + 14)(x^2 - 4x - 5x + 20) - 72`
`= (x^2 - 9x + 14)(x^2 - 9x + 20) - 72`
Đặt `x^2 - 9x + 17 = a`, ta được:
`B = (a - 3)(a + 3) - 72`
`= a^2 - 9 - 72`
`= a^2 - 81 = a^2 - 9^2`
`= (a - 9)(a + 9)`
`= (x^2 - 9x + 17 - 9)(x^2 - 9x + 17 + 9)`
`= (x^2 - 9x + 8)(x^2 - 9x + 26)`
`= (x^2 - x - 8x + 8)(x^2 - 9x + 26)`
`= [x(x - 1) - 8(x - 1)](x^2 - 9x + 26)`
`= (x - 1)(x - 8)(x^2 - 9x + 26)`
`C = (x + 2)(x + 3)(x + 8)(x + 12) - 4x^2`
`= [(x + 2)(x +12)][(x + 3)(x + 8)] - 4x^2`
`= (x^2 + 14x + 24)(x^2 + 11x + 24) - 4x^2`
Đặt `x^2 + 11x + 24 = a`, ta được:
`C = (a + 3x).a - 4x^2`
`= a^2 + 3ax - 4x^2`
`= a^2 - ax + 4ax - 4x^2`
`= a(a - x) + 4x(a - x)`
`= (a - x)(a + 4x)`
`= (x^2 + 11x + 24 - x)(x^2 + 11x + 24 + 4x)`
`= (x^2 + 10x + 24)(x^2 + 15x + 24)`
`= (x^2 + 4x + 6x + 24)(x^2 + 15x + 24)`
`= [x(x+4) + 6(x + 4)](x^2 + 15x + 24)`
`=(x + 4)(x + 6)(x^2 + 15x + 24)`
`D = (x - 1)(x + 2)(x + 3)(x - 6) + 32x^2`
`= [(x - 1)(x - 6)][(x + 2)(x + 3)] + 32x^2`
`= (x^2 - 7x + 6)(x^2 + 5x + 6) + 32x^2`
Đặt `x^2 - x + 6 = a`, ta được:
`D = (a - 6x)(a + 6x) + 32x^2`
`= a^2 - (6x)^2 + 32x^2`
`= a^2 - 36x^2 + 32x^2 = a^2 - 4x^2`
`= a^2 - (2x)^2`
`= (a - 2x)(a + 2x)`
`= (x^2 - x + 6 - 2x)(x^2 - x + 6 + 2x)`
`= (x^2 - 3x + 6)(x^2 + x + 6)`
`E = x^4 - 10x^3 + 26x^2 - 10x + 1`
`= x^2 (x^2 - 10x + 26 - 10/x + 1/x^2)`
`= x^2 [(x^2 + 1/x^2) + 26 - 10(x + 1/x)]`
Đặt `x + 1/x = t ⇒ t^2 = x^2 + 2 + 1/x^2`
`⇒ x^2 + 1/x^2 = t^2 - 2`
Khi đó đa thức trở thành:
`E = x^2 (t^2 - 2 + 26 - 10t)`
`= x^2 (t^2 - 10t + 24)`
`= x^2 (t^2 - 4t - 6t +24)`
`= x^2 [t(t - 4) - 6(t - 4)]`
`= x^2 (t - 4)(t - 6)`
`= x^2 (x + 1/x - 4)(x + 1/x - 6)`
`= [x(x + 1/x - 4)][x(x + 1/x - 6)]`
`= (x^2 - 4x + 1)(x^2 - 6x + 1)`
`F = 2x^4 - 5x^3 - 27x^2 + 25x + 50`
`= 2x^4 - 7x^3 - 10x^2 + 2x^3 - 7x^2 - 10x - 10x^2 + 35x + 50`
`= x^2 (2x^2 - 7x - 10) + x(2x^2 - 7x - 10) - 5(2x^2 - 7x - 10)`
`= (2x^2 - 7x - 10)(x^2 + x - 5)`
`G = x^4 - 10x^3 - 15x^2 + 20x + 4`
`= x^4 - x^3 - 9x^3 + 9x^2 - 24x^2 + 24x - 4x + 4`
`= (x^4 - x^3) - (9x^3 - 9x^2) - (24x^2 - 24x) - (4x - 4)`
`= x^3 (x - 1) - 9x^2 (x - 1) - 24x(x - 1) - 4(x - 1)`
`= (x - 1)(x^3 - 9x^2 - 24x - 4)`
`= (x - 1)(x^3 + 2x^2 - 11x^2 - 22x - 2x - 4)`
`= (x - 1)[x^2 (x + 2) - 11x(x + 2) - 2(x + 2)]`
`= (x - 1)(x + 2)(x^2 - 11x - 2)`
