단순화 (x+y)^3(3x-2y)^4+2(x+y)^4(3x-2y)^3

해법

단순화

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

135 x^{7} + 135 x^{6} y - 225 x^{5} y^{2} - 175 x^{4} y^{3} + 150 x^{3} y^{4} + 60 x^{2} y^{5} - 40 x y^{6}

솔루션 단계

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

(x + y)^{3} : x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}

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

(3 x - 2 y)^{4} : 81 x^{4} - 216 x^{3} y + 216 x^{2} y^{2} - 96 x y^{3} + 16 y^{4}

= (x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}) (81 x^{4} - 216 x^{3} y + 216 x^{2} y^{2} - 96 x y^{3} + 16 y^{4}) + 2 (x + y)^{4} (3 x - 2 y)^{3}

(x + y)^{4} : x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}

= (x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}) (81 x^{4} - 216 x^{3} y + 216 x^{2} y^{2} - 96 x y^{3} + 16 y^{4}) + 2 (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}) (3 x - 2 y)^{3}

(3 x - 2 y)^{3} : 27 x^{3} - 54 x^{2} y + 36 x y^{2} - 8 y^{3}

= (x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}) (81 x^{4} - 216 x^{3} y + 216 x^{2} y^{2} - 96 x y^{3} + 16 y^{4}) + 2 (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}) (27 x^{3} - 54 x^{2} y + 36 x y^{2} - 8 y^{3})

(x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}) (81 x^{4} - 216 x^{3} y + 216 x^{2} y^{2} - 96 x y^{3} + 16 y^{4})

확대한다:

81 x^{7} + 27 x^{6} y - 189 x^{5} y^{2} - 15 x^{4} y^{3} + 160 x^{3} y^{4} - 24 x^{2} y^{5} - 48 x y^{6} + 16 y^{7}

= 81 x^{7} + 27 x^{6} y - 189 x^{5} y^{2} - 15 x^{4} y^{3} + 160 x^{3} y^{4} - 24 x^{2} y^{5} - 48 x y^{6} + 16 y^{7} + 2 (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}) (27 x^{3} - 54 x^{2} y + 36 x y^{2} - 8 y^{3})

2 (x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}) (27 x^{3} - 54 x^{2} y + 36 x y^{2} - 8 y^{3})

확대한다:

54 x^{7} + 108 x^{6} y - 36 x^{5} y^{2} - 160 x^{4} y^{3} - 10 x^{3} y^{4} + 84 x^{2} y^{5} + 8 x y^{6} - 16 y^{7}

= 81 x^{7} + 27 x^{6} y - 189 x^{5} y^{2} - 15 x^{4} y^{3} + 160 x^{3} y^{4} - 24 x^{2} y^{5} - 48 x y^{6} + 16 y^{7} + 54 x^{7} + 108 x^{6} y - 36 x^{5} y^{2} - 160 x^{4} y^{3} - 10 x^{3} y^{4} + 84 x^{2} y^{5} + 8 x y^{6} - 16 y^{7}

81 x^{7} + 27 x^{6} y - 189 x^{5} y^{2} - 15 x^{4} y^{3} + 160 x^{3} y^{4} - 24 x^{2} y^{5} - 48 x y^{6} + 16 y^{7} + 54 x^{7} + 108 x^{6} y - 36 x^{5} y^{2} - 160 x^{4} y^{3} - 10 x^{3} y^{4} + 84 x^{2} y^{5} + 8 x y^{6} - 16 y^{7}

단순화하세요:

135 x^{7} + 135 x^{6} y - 225 x^{5} y^{2} - 175 x^{4} y^{3} + 150 x^{3} y^{4} + 60 x^{2} y^{5} - 40 x y^{6}

= 135 x^{7} + 135 x^{6} y - 225 x^{5} y^{2} - 175 x^{4} y^{3} + 150 x^{3} y^{4} + 60 x^{2} y^{5} - 40 x y^{6}