化简 (x-1-2i)^3(x-1+2i)

解答

化简

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

(x^{4} - 4 x^{3} + 6 x^{2} - 4 x - 15) + (20 - 28 x + 12 x^{2} - 4 x^{3}) i

求解步骤

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

使用分配律:

m (a + b + c) = ma + mb + m c

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

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

化简

(x - 1 - 2 i)^{3} x + (x - 1 - 2 i)^{3} (- 1) + (x - 1 - 2 i)^{3} \cdot 2 i : 20 i + x^{4} - 4 x^{3} + 6 x^{2} - 28 x i + 12 x^{2} i - 4 x^{3} i - 4 x - 15

= 20 i + x^{4} - 4 x^{3} + 6 x^{2} - 28 x i + 12 x^{2} i - 4 x^{3} i - 4 x - 15

Rewrite in standard complex form:

(x^{4} - 4 x^{3} + 6 x^{2} - 4 x - 15) + (20 - 28 x + 12 x^{2} - 4 x^{3}) i

= (x^{4} - 4 x^{3} + 6 x^{2} - 4 x - 15) + (20 - 28 x + 12 x^{2} - 4 x^{3}) i

s im pl i f y (3 + \frac{3 i}{n})^{2}

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