Question

Phân tích các đa thức sau thành nhân tử:

a) \(\left(x^2-x+2\right)^4-3x^2.\left(x^2-x+2\right)^2+2x^4\)

b) \(3.\left(-x^2+2x+3\right)^4-26x^2.\left(-x^2+2x+3\right)-9x^4\)

c) \(\left(x^2-x-1\right)^4+7x^2.\left(x^2-x+1\right)^2+12x^4\)

Làm theo phương pháp đặt ẩn phụ nhé m.n !

Answer 1

a: Đặt \(\left(x^2-x+2\right)^2=a\)

Biểu thức trở thành \(a^2-3x^2a+2x^4\)

\(=a^2-x^2a-2x^2a+2x^4\)

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

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

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

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

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

\(=\left(x-2\right)\left(x+2\right)\left(x-1\right)\)

b:

Sửa đề: \(3\left(-x^2+2x+3\right)^4-26x^2\left(-x^2+2x+3\right)^2-9x^4\)

Đặt \(\left(-x^2+2x+3\right)^2=a\)

BT trở thành \(3a^2-26x^2a-9x^4\)

\(=3a^2-27x^2a+x^2a-9x^4\)

\(=3a\left(a-9x^2\right)+x^2\left(a-9x^2\right)\)

\(=\left(a-9x^2\right)\left(3a+x^2\right)\)

\(=\left[\left(x^2-2x-3\right)^2-9x^2\right]\left[x^2+\left(x^2-2x-3\right)^2\right]\)

\(=\left(x^2-2x-3-3x\right)\left(x^2-2x-3+3x\right)\left[x^2+\left(x^2-2x-3\right)^2\right]\)

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