Question

Phân tích đa thức thành nhân tử

Câu 1: a³+a²b-ab²-b³

Câu 2: a(b³-c³)+(c³-a³)+c(a³-b³)

Câu 3: a⁴+b⁴+c⁴-2a²b²-2b²c²-2a²c²

Answer 1

Câu 1:

\(a^3+a^2b-ab^2-b^3\)

\(=a^2\left(a+b\right)-b^2\left(a+b\right)\)

\(=\left(a+b\right)\left(a^2-b^2\right)\)

\(=\left(a+b\right)\left(a-b\right)\left(a+b\right)\)

\(=\left(a+b\right)^2\left(a-b\right)\)

Answer 2

Câu 2:

\(a\left(b^3-c^3\right)+b\left(c^3-a^3\right)+c\left(a^3-b^3\right)\)

\(=a\left(b^3-c^3\right)+bc^3-a^3b+a^3c-b^3c\)

\(=a\left(b-c\right)\left(b^2+bc+c^2\right)-a^3\left(b-c\right)-bc\left(b-c\right)\left(b+c\right)\)

\(=\left(b-c\right)\left(ab^2+abc+c^2a-a^3-b^2c-bc^2\right)\)

\(=\left(b-c\right)\left[a\left(c-a\right)\left(c+a\right)-b^2\left(c-a\right)-bc\left(c-a\right)\right]\)

\(=\left(b-c\right)\left(c-a\right)\left(ca+a^2-b^2-bc\right)\)

\(=\left(b-c\right)\left(c-a\right)\left[\left(a-b\right)\left(a+b\right)+c\left(a-b\right)\right]\)

\(=\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+b+c\right)\)

Answer 3

Câu 3:

\(a^4+b^4+c^4-2a^2b^2-2b^2c^2-2c^2a^2\)

\(=\left(a^4-2a^2b^2+b^4\right)+\left(2c^2a^2-2b^2c^2\right)+c^4-4c^2a^2\)

\(=\left(a^2-b^2\right)^2+2c^2\left(a^2-b^2\right)+c^4-4c^2a^2\)

\(=\left(a^2-b^2+c^2\right)-\left(2ca\right)^2\)

\(=\left(a^2-b^2+c^2-2ca\right)\left(a^2-b^2+c^2+2ca\right)\)