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Popular Algebra >

expand (2y-5)^7

Solution

expand

(2 y - 5)^{7}

Solution

128 y^{7} - 2240 y^{6} + 16800 y^{5} - 70000 y^{4} + 175000 y^{3} - 262500 y^{2} + 218750 y - 78125

Solution steps

(2 y - 5)^{7}

Apply binomial theorem:

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

a = 2 y, b = - 5

= i = 0 \sum 7 (i 7) (2 y)^{(7 - i)} (- 5)^{i}

Expand summation

= \frac{7 !}{0 ! ( 7 - 0 ) !} (2 y)^{7} (- 5)^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} (2 y)^{6} (- 5)^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} (2 y)^{5} (- 5)^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} (2 y)^{4} (- 5)^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} (2 y)^{3} (- 5)^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} (2 y)^{2} (- 5)^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} (2 y)^{1} (- 5)^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} (2 y)^{0} (- 5)^{7}

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} (2 y)^{7} (- 5)^{0} : 128 y^{7}

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} (2 y)^{6} (- 5)^{1} : - 2240 y^{6}

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} (2 y)^{5} (- 5)^{2} : 16800 y^{5}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} (2 y)^{4} (- 5)^{3} : - 70000 y^{4}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} (2 y)^{3} (- 5)^{4} : 175000 y^{3}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} (2 y)^{2} (- 5)^{5} : - 262500 y^{2}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} (2 y)^{1} (- 5)^{6} : 218750 y

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} (2 y)^{0} (- 5)^{7} : - 78125

= 128 y^{7} - 2240 y^{6} + 16800 y^{5} - 70000 y^{4} + 175000 y^{3} - 262500 y^{2} + 218750 y - 78125

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