因式分解 sin^{25}(x)+cos^{25}(x)

解答

因式分解

sin^{25} (x) + cos^{25} (x)

(sin (x) + cos (x)) (sin^{24} (x) - sin^{23} (x) cos (x) + sin^{22} (x) cos^{2} (x) - sin^{21} (x) cos^{3} (x) + sin^{20} (x) cos^{4} (x) - sin^{19} (x) cos^{5} (x) + sin^{18} (x) cos^{6} (x) - sin^{17} (x) cos^{7} (x) + sin^{16} (x) cos^{8} (x) - sin^{15} (x) cos^{9} (x) + sin^{14} (x) cos^{10} (x) - sin^{13} (x) cos^{11} (x) + sin^{12} (x) cos^{12} (x) - cos^{13} (x) sin^{11} (x) + cos^{14} (x) sin^{10} (x) - cos^{15} (x) sin^{9} (x) + cos^{16} (x) sin^{8} (x) - cos^{17} (x) sin^{7} (x) + cos^{18} (x) sin^{6} (x) - cos^{19} (x) sin^{5} (x) + cos^{20} (x) sin^{4} (x) - cos^{21} (x) sin^{3} (x) + cos^{22} (x) sin^{2} (x) - cos^{23} (x) sin (x) + cos^{24} (x))

求解步骤

sin^{25} (x) + cos^{25} (x)

使用因式分解法则

x^{n} + y^{n} = (x + y) (x^{n - 1} - x^{n - 2} y + \dots - x y^{n - 2} + y^{n - 1})

n is odd

sin^{25} (x) + cos^{25} (x) = (sin (x) + cos (x)) (sin^{24} (x) - sin^{23} (x) cos (x) + sin^{22} (x) cos^{2} (x) - sin^{21} (x) cos^{3} (x) + sin^{20} (x) cos^{4} (x) - sin^{19} (x) cos^{5} (x) + sin^{18} (x) cos^{6} (x) - sin^{17} (x) cos^{7} (x) + sin^{16} (x) cos^{8} (x) - sin^{15} (x) cos^{9} (x) + sin^{14} (x) cos^{10} (x) - sin^{13} (x) cos^{11} (x) + sin^{12} (x) cos^{12} (x) - sin^{11} (x) cos^{13} (x) + sin^{10} (x) cos^{14} (x) - sin^{9} (x) cos^{15} (x) + sin^{8} (x) cos^{16} (x) - sin^{7} (x) cos^{17} (x) + sin^{6} (x) cos^{18} (x) - sin^{5} (x) cos^{19} (x) + sin^{4} (x) cos^{20} (x) - sin^{3} (x) cos^{21} (x) + sin^{2} (x) cos^{22} (x) - sin (x) cos^{23} (x) + cos^{24} (x))

= (sin (x) + cos (x)) (sin^{24} (x) - sin^{23} (x) cos (x) + sin^{22} (x) cos^{2} (x) - sin^{21} (x) cos^{3} (x) + sin^{20} (x) cos^{4} (x) - sin^{19} (x) cos^{5} (x) + sin^{18} (x) cos^{6} (x) - sin^{17} (x) cos^{7} (x) + sin^{16} (x) cos^{8} (x) - sin^{15} (x) cos^{9} (x) + sin^{14} (x) cos^{10} (x) - sin^{13} (x) cos^{11} (x) + sin^{12} (x) cos^{12} (x) - sin^{11} (x) cos^{13} (x) + sin^{10} (x) cos^{14} (x) - sin^{9} (x) cos^{15} (x) + sin^{8} (x) cos^{16} (x) - sin^{7} (x) cos^{17} (x) + sin^{6} (x) cos^{18} (x) - sin^{5} (x) cos^{19} (x) + sin^{4} (x) cos^{20} (x) - sin^{3} (x) cos^{21} (x) + sin^{2} (x) cos^{22} (x) - sin (x) cos^{23} (x) + cos^{24} (x))

f a c t or 6^{3} - 13 x - 5

f (x) = (sin (5 x))^{2}

f a c t or (x - 3)^{3} + (3 x - 2)^{2}

f a c t or y^{2} d x - x y d y

s im pl i f y x^{2} + 2 x y + (y)^{2} - 4 x - 4 y