Factorization show steps

Put u = x²

x⁴ + 4x² + 3
= (x²)² + 4(x²) + 3
= u² + 4u + 3
= (u + 1)(u + 3)
= (x² + 1)(x² + 3)


Put f(x) = x⁴ - 4x³ + 4x² - 4x + 3

f(1) = (1)⁴ - 4(1)³ + 4(1)² - 4(1) + 3 = 0
f(3) = (3)⁴ - 4(3)³ + 4(3)² - 4(3) + 3 = 0
Hence, x - 1 and x - 3 are the factors of f(x).

(x - 1)(x - 3) = x² - 4x + 3
Using long division, (x⁴ - 4x³ + 4x² - 4x + 3) ÷ (x² - 4x + 3) = (x² + 1)

Hence, (x⁴ - 4x³ + 4x² - 4x + 3)
= (x² - 4x + 3)(x² + 1)
= (x - 1)(x -3)(x² + 1)

2011-11-18 23:33:19 補充：
Alternative method for Q.2 :

x⁴ - 4x³ + 4x² - 4x + 3
= (x⁴ - 3x³) - (x³ - 3x²) + (x² - 3x) - (x - 3)
= x³(x - 3) - x²(x - 3) + x(x - 3) - (x - 3)
= (x³ - x² + x - 1)(x - 3)
= [(x³ - x²) + (x - 1)] (x - 3)
= [x²(x - 1) + (x - 1)] (x - 3)
= (x² + 1)(x - 1)(x - 3) ...... (answer)

x^4 - 4x^3 +4x^2 -4x + 3
=(x^4 +4x^2 +3) -(4x^3 +4x)
=(x^2 +1)(x^2 +3) - 4x(x^2 +1)
=(x^2 +1)(x^2 -4x+3)
=(x^2 +1)(x-3)(x-1)