Question

1) rút gọn biểu thức sau

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

2) phân tích các đa thức sau thành nhân tử

a) \(x^4-3x^3-x+3\)

b) \(x^3+27+\left(x+3\right)\left(x-9\right)\)

Answer 1

1)Ta có: \(\left(x+1\right)\left(x^2-x+1\right)-\left(x-1\right)\left(x^2+x+1\right)\)

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

2)

a) Ta có: \(x^4-3x^3-x+3=\left(x^4-3x^3\right)-\left(x-3\right)=x^3\left(x-3\right)-\left(x-3\right)=\left(x-3\right)\left(x^3-1\right)=\left(x-3\right)\left(x-1\right)\left(x^2+x+1\right)\)b) Ta có: \(x^3+27+\left(x+3\right)\left(x-9\right)\)

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

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

Answer 2

1)(x+1)(x^2-x+1)-(x-1)(x^2+x+1)

=x^3+1-(x^3-1)

=x^3+1-x^3+1

=2

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

=x^3(x-3)-(x-3)

=(x^3-1)(x-3)

=(x-1)(x^2+x+1)(x-3)

b)x^3+27+(x+3)(x-9)

=x^3+27+x^2-9x+3x-27

=x^3+x^2-6x

=x(x^2+3x-2x-6)

=x[x(x+3)-2(x+3)]

=x(x-2)(x+3)