化简 x^3(x-2+i)(x-2+i)(3x+2)

解答

化简

x^{3} (x - 2 + i) (x - 2 + i) (3 x + 2)

x^{3} (3 x + 2) (x^{2} - 4 x + 3) + x^{3} (3 x + 2) (2 x - 4) i

求解步骤

x^{3} (x - 2 + i) (x - 2 + i) (3 x + 2)

使用复数算数法则:

(a + bi) (c + d i) = (a c - b d) + (a d + b c) i

= x^{3} (((x - 2) (x - 2) - 1 \cdot 1) + ((x - 2) \cdot 1 + 1 \cdot (x - 2)) i) (3 x + 2)

化简

((x - 2) (x - 2) - 1 \cdot 1) + ((x - 2) \cdot 1 + 1 \cdot (x - 2)) i : (x^{2} - 4 x + 3) + (2 x - 4) i

= x^{3} ((x^{2} - 4 x + 3) + (2 x - 4) i) (3 x + 2)

打开括号

= x^{3} (3 x + 2) (x^{2} - 4 x + 3) + x^{3} (3 x + 2) (2 x - 4) i

e x p an d (- \frac{x ^{3}}{3} + \frac{x ^{2}}{2}) (e^{2 x})

e x p an d 8 r + 12 p - 7 - 3 p

e x p an d π (2 x + 1)^{2} (2 x - 1)

e x p an d \frac{2 e ^{x^{2}} ( x ^{2} - 1 )}{x ^{3}}

e x p an d (5 x^{\frac{1}{2}} + 3) (5 x^{\frac{1}{2}} - 3)