化简 (r-1)^3(r+4-2i)^4r

解答

化简

(r - 1)^{3} (r + 4 - 2 i)^{4} r

(r^{8} + 13 r^{7} + 27 r^{6} - 105 r^{5} - 104 r^{4} + 456 r^{3} - 400 r^{2} + 112 r) + (392 r^{4} - 88 r^{5} - 8 r^{7} - 72 r^{6} + 192 r^{3} - 800 r^{2} + 384 r) i

求解步骤

(r - 1)^{3} (r + 4 - 2 i)^{4} r

= r (r - 1)^{3} (r + 4 - 2 i)^{4}

展开

(r - 1)^{3} : r^{3} - 3 r^{2} + 3 r - 1

= r (r^{3} - 3 r^{2} + 3 r - 1) (r + 4 - 2 i)^{4}

= r (r + 4 - 2 i)^{4} (r^{3} - 3 r^{2} + 3 r - 1)

打开括号

= r (r + 4 - 2 i)^{4} r^{3} + r (r + 4 - 2 i)^{4} (- 3 r^{2}) + r (r + 4 - 2 i)^{4} \cdot 3 r + r (r + 4 - 2 i)^{4} (- 1)

化简

r (r + 4 - 2 i)^{4} r^{3} + r (r + 4 - 2 i)^{4} (- 3 r^{2}) + r (r + 4 - 2 i)^{4} \cdot 3 r + r (r + 4 - 2 i)^{4} (- 1) : r^{8} + 13 r^{7} + 27 r^{6} - 105 r^{5} + 392 r^{4} i - 88 r^{5} i - 8 r^{7} i - 72 r^{6} i - 104 r^{4} + 192 r^{3} i + 456 r^{3} - 800 r^{2} i - 400 r^{2} + 384 r i + 112 r

= r^{8} + 13 r^{7} + 27 r^{6} - 105 r^{5} + 392 r^{4} i - 88 r^{5} i - 8 r^{7} i - 72 r^{6} i - 104 r^{4} + 192 r^{3} i + 456 r^{3} - 800 r^{2} i - 400 r^{2} + 384 r i + 112 r

Rewrite in standard complex form:

(r^{8} + 13 r^{7} + 27 r^{6} - 105 r^{5} - 104 r^{4} + 456 r^{3} - 400 r^{2} + 112 r) + (392 r^{4} - 88 r^{5} - 8 r^{7} - 72 r^{6} + 192 r^{3} - 800 r^{2} + 384 r) i

= (r^{8} + 13 r^{7} + 27 r^{6} - 105 r^{5} - 104 r^{4} + 456 r^{3} - 400 r^{2} + 112 r) + (392 r^{4} - 88 r^{5} - 8 r^{7} - 72 r^{6} + 192 r^{3} - 800 r^{2} + 384 r) i

e x p an d (2 y^{2} + 2 y + 4 x^{2}) d x + (2 x y + x) d y

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

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

e x p an d 3 l e f t (2 a + 5 a^{2}) + (2 a - a^{2})

e x p an d e^{e^{x} (x - 1)}