Question

Phân tích các đa thức sau thành nhân tử:

a) \(\text{x}^{\text{4}}+3\text{x}^{\text{3}}-7\text{x}^{\text{2}}-27x-18\)

b) \(\text{x}^{\text{4}}+3\text{x}^{\text{3}}+3\text{x}^{\text{2}}+3x+2\)

Answer 1

a) x⁴ + 3x³ - 7x² - 27x - 18

= x⁴ + x³ + 2x³ + 2x² - 9x² - 9x - 18x - 18

= x³ . (x + 1) + 2x² . (x + 1) - 9x . (x + 1) - 18(x + 1)

= (x + 1)(x³ + 2x² - 9x - 18)

= (x + 1)[x² .(x + 2) - 9.(x + 2)]

= (x + 1)(x + 2)(x² - 3²)

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

b) x⁴ + 3x³ + 3x² + 3x + 2

= x⁴ + x³ + 2x³ + 2x² + x² + x + 2x + 2

= x³ (x + 1) + 2x² . (x + 1) + x(x + 1) + 2(x + 1)

= (x + 1)(x³ + 2x² + x + 2)

= (x + 1)[x² .(x + 2) + (x + 2)]

= (x + 1)(x + 2)(x² + 1)

Answer 2

\(x^4+3x^3-7x^2-27x-18\)

\(=\left(x^4+x^3\right)+\left(2x^3+2x^2\right)-\left(9x^2+9x\right)-\left(18x-18\right)\)

\(=x^3\left(x+1\right)+2x^2\left(x+1\right)-9x\left(x+1\right)-18\left(x+1\right)\)

\(=\left(x+1\right)\left(x^3+2x^2-9x-18\right)\)

\(=\left(x+1\right)\left[\left(x^3-3x^2\right)+\left(5x^2-15x\right)+\left(6x-18\right)\right]\)

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

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

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

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