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Popular Algebra >

expand (3x+2y)^{17}

Solution

expand

(3 x + 2 y)^{17}

Solution

129140163 x^{17} + 1463588514 x^{16} y + 7805805408 x^{15} y^{2} + 26019351360 x^{14} y^{3} + 60711819840 x^{13} y^{4} + 105233821056 x^{12} y^{5} + 140311761408 x^{11} y^{6} + 146993273856 x^{10} y^{7} + 122494394880 x^{9} y^{8} + 81662929920 x^{8} y^{9} + 43553562624 x^{7} y^{10} + 18477268992 x^{6} y^{11} + 6159089664 x^{5} y^{12} + 1579253760 x^{4} y^{13} + 300810240 x^{3} y^{14} + 40108032 x^{2} y^{15} + 3342336 x y^{16} + 131072 y^{17}

Solution steps

(3 x + 2 y)^{17}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 3 x, b = 2 y

= i = 0 \sum 17 (i 17) (3 x)^{(17 - i)} (2 y)^{i}

Expand summation

= \frac{17 !}{0 ! ( 17 - 0 ) !} (3 x)^{17} (2 y)^{0} + \frac{17 !}{1 ! ( 17 - 1 ) !} (3 x)^{16} (2 y)^{1} + \frac{17 !}{2 ! ( 17 - 2 ) !} (3 x)^{15} (2 y)^{2} + \frac{17 !}{3 ! ( 17 - 3 ) !} (3 x)^{14} (2 y)^{3} + \frac{17 !}{4 ! ( 17 - 4 ) !} (3 x)^{13} (2 y)^{4} + \frac{17 !}{5 ! ( 17 - 5 ) !} (3 x)^{12} (2 y)^{5} + \frac{17 !}{6 ! ( 17 - 6 ) !} (3 x)^{11} (2 y)^{6} + \frac{17 !}{7 ! ( 17 - 7 ) !} (3 x)^{10} (2 y)^{7} + \frac{17 !}{8 ! ( 17 - 8 ) !} (3 x)^{9} (2 y)^{8} + \frac{17 !}{9 ! ( 17 - 9 ) !} (3 x)^{8} (2 y)^{9} + \frac{17 !}{10 ! ( 17 - 10 ) !} (3 x)^{7} (2 y)^{10} + \frac{17 !}{11 ! ( 17 - 11 ) !} (3 x)^{6} (2 y)^{11} + \frac{17 !}{12 ! ( 17 - 12 ) !} (3 x)^{5} (2 y)^{12} + \frac{17 !}{13 ! ( 17 - 13 ) !} (3 x)^{4} (2 y)^{13} + \frac{17 !}{14 ! ( 17 - 14 ) !} (3 x)^{3} (2 y)^{14} + \frac{17 !}{15 ! ( 17 - 15 ) !} (3 x)^{2} (2 y)^{15} + \frac{17 !}{16 ! ( 17 - 16 ) !} (3 x)^{1} (2 y)^{16} + \frac{17 !}{17 ! ( 17 - 17 ) !} (3 x)^{0} (2 y)^{17}

Simplify

\frac{17 !}{0 ! ( 17 - 0 ) !} (3 x)^{17} (2 y)^{0} : 129140163 x^{17}

Simplify

\frac{17 !}{1 ! ( 17 - 1 ) !} (3 x)^{16} (2 y)^{1} : 1463588514 x^{16} y

Simplify

\frac{17 !}{2 ! ( 17 - 2 ) !} (3 x)^{15} (2 y)^{2} : 7805805408 x^{15} y^{2}

Simplify

\frac{17 !}{3 ! ( 17 - 3 ) !} (3 x)^{14} (2 y)^{3} : 26019351360 x^{14} y^{3}

Simplify

\frac{17 !}{4 ! ( 17 - 4 ) !} (3 x)^{13} (2 y)^{4} : 60711819840 x^{13} y^{4}

Simplify

\frac{17 !}{5 ! ( 17 - 5 ) !} (3 x)^{12} (2 y)^{5} : 105233821056 x^{12} y^{5}

Simplify

\frac{17 !}{6 ! ( 17 - 6 ) !} (3 x)^{11} (2 y)^{6} : 140311761408 x^{11} y^{6}

Simplify

\frac{17 !}{7 ! ( 17 - 7 ) !} (3 x)^{10} (2 y)^{7} : 146993273856 x^{10} y^{7}

Simplify

\frac{17 !}{8 ! ( 17 - 8 ) !} (3 x)^{9} (2 y)^{8} : 122494394880 x^{9} y^{8}

Simplify

\frac{17 !}{9 ! ( 17 - 9 ) !} (3 x)^{8} (2 y)^{9} : 81662929920 x^{8} y^{9}

Simplify

\frac{17 !}{10 ! ( 17 - 10 ) !} (3 x)^{7} (2 y)^{10} : 43553562624 x^{7} y^{10}

Simplify

\frac{17 !}{11 ! ( 17 - 11 ) !} (3 x)^{6} (2 y)^{11} : 18477268992 x^{6} y^{11}

Simplify

\frac{17 !}{12 ! ( 17 - 12 ) !} (3 x)^{5} (2 y)^{12} : 6159089664 x^{5} y^{12}

Simplify

\frac{17 !}{13 ! ( 17 - 13 ) !} (3 x)^{4} (2 y)^{13} : 1579253760 x^{4} y^{13}

Simplify

\frac{17 !}{14 ! ( 17 - 14 ) !} (3 x)^{3} (2 y)^{14} : 300810240 x^{3} y^{14}

Simplify

\frac{17 !}{15 ! ( 17 - 15 ) !} (3 x)^{2} (2 y)^{15} : 40108032 x^{2} y^{15}

Simplify

\frac{17 !}{16 ! ( 17 - 16 ) !} (3 x)^{1} (2 y)^{16} : 3342336 x y^{16}

Simplify

\frac{17 !}{17 ! ( 17 - 17 ) !} (3 x)^{0} (2 y)^{17} : 131072 y^{17}

= 129140163 x^{17} + 1463588514 x^{16} y + 7805805408 x^{15} y^{2} + 26019351360 x^{14} y^{3} + 60711819840 x^{13} y^{4} + 105233821056 x^{12} y^{5} + 140311761408 x^{11} y^{6} + 146993273856 x^{10} y^{7} + 122494394880 x^{9} y^{8} + 81662929920 x^{8} y^{9} + 43553562624 x^{7} y^{10} + 18477268992 x^{6} y^{11} + 6159089664 x^{5} y^{12} + 1579253760 x^{4} y^{13} + 300810240 x^{3} y^{14} + 40108032 x^{2} y^{15} + 3342336 x y^{16} + 131072 y^{17}

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