Question

Phân tích đa thức thành nhân tử:

a) (xy + 1)² - (x + y)²           

b) (x + y)³ - (x - y)³

c) 3x⁴y² + 3x³y² + 3xy² + 3y²

d) 4(x² - y²) - 8(x - ay) - 4(a² - 1)

Answer 1

a) \(\left(xy+1\right)^2-\left(x+y\right)\)

\(=\left(xy+1-x-y\right)\left(xy+1+x+y\right)\)

\(=\left[x\left(y-1\right)-\left(y-1\right)\right]\left[x\left(y+1\right)+\left(y+1\right)\right]\)

\(=\left(x-1\right)\left(y-1\right)\left(x+1\right)\left(y+1\right)\)

Answer 2

b) \(\left(x+y\right)^3-\left(x-y\right)^3\)

\(=\left(x+y-x+y\right)\left[\left(x+y\right)^2+\left(x-y\right)\left(x+y\right)+\left(x-y\right)^2\right]\)

\(=2y\left(x^2+2xy+y^2+x^2-y^2+x^2-2xy+y^2\right)\)

\(=2y\left(3x^2+y^2\right)\)

Answer 3

c) \(3x^4y^2+3x^3y^2+3xy^2+3y^2\)

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

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

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

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

Answer 4

d) \(4\left(x^2-y^2\right)-8\left(x-ay\right)-4\left(a^2-1\right)\)

\(=4x^2-4y^2-8x+8ay-4a^2+4\)

\(=\left(4x^2-8x+4\right)-\left(4a^2-8ay+4y^2\right)\)

\(=\left(2x-2\right)^2-\left(2a-2y\right)^2\)

\(=4\left(x-1-a+y\right)\left(x-1+a-y\right)\)