Question

Phân tích các đa thức sau thành nhân tử:

a) x^3 - 5x^2 + 8x - 4

b) x^3 - 7x + 6

c) x^3 + 2x^2 + 4x + 3

d) 2x^3 - x^2 - 5x - 2

Answer 1

a)

\(x^3-5x^2+8x-4\\ =\left(x^3-x^2\right)-\left(4x^2-4x\right)+\left(4x-4\right)\\ =x^2\left(x-1\right)-4x\left(x-1\right)+4\left(x-1\right)\\ =\left(x-1\right)\left(x^2-4x+4\right)\\ =\left(x-1\right)\left(x-2\right)^2\)

b)

\(x^3-7x+6\\ =\left(x^3-2x^2\right)+\left(2x^2-4x\right)-\left(3x-6\right)\\ =x^2\left(x-2\right)+2x\left(x-2\right)-3\left(x-2\right)\\ =\left(x-2\right)\left(x^2+2x-3\right)\)

c)

\(x^3+2x^2+4x+3\\ =\left(x^3+x^2\right)+\left(x^2+x\right)+\left(3x+3\right)\\ =x^2\left(x+1\right)+x\left(x+1\right)+3\left(x+1\right)\\ =\left(x+1\right)\left(x^2+x+3\right)\)

d)

\(2x^3-x^2-5x-2\\ =\left(2x^3-4x^2\right)+\left(3x^2-6x\right)+\left(x-2\right)\\ =2x^2\left(x-2\right)+3x\left(x-2\right)+\left(x-2\right)\\ =\left(x-2\right)\left(2x^2+3x+1\right)\\ =\left(x-2\right)\left[2x\left(x+1\right)+\left(x+1\right)\right]\\ =\left(x-2\right)\left(x+1\right)\left(2x+1\right)\)