What is expand (-2x+1/2)^8 ?

The solution to expand (-2x+1/2)^8 is 256x^8-512x^7+448x^6-224x^5+70x^4-14x^3+(7x^2)/4-x/8+1/256

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expand (-2x+1/2)^8

expand

(- 2 x + \frac{1}{2})^{8}

256 x^{8} - 512 x^{7} + 448 x^{6} - 224 x^{5} + 70 x^{4} - 14 x^{3} + \frac{7 x ^{2}}{4} - \frac{x}{8} + \frac{1}{256}

Solution steps

(- 2 x + \frac{1}{2})^{8}

Apply binomial theorem:

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

a = - 2 x, b = \frac{1}{2}

= i = 0 \sum 8 (i 8) (- 2 x)^{(8 - i)} (\frac{1}{2})^{i}

Expand summation

= \frac{8 !}{0 ! ( 8 - 0 ) !} (- 2 x)^{8} (\frac{1}{2})^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} (- 2 x)^{7} (\frac{1}{2})^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} (- 2 x)^{6} (\frac{1}{2})^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} (- 2 x)^{5} (\frac{1}{2})^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} (- 2 x)^{4} (\frac{1}{2})^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} (- 2 x)^{3} (\frac{1}{2})^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} (- 2 x)^{2} (\frac{1}{2})^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} (- 2 x)^{1} (\frac{1}{2})^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} (- 2 x)^{0} (\frac{1}{2})^{8}

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} (- 2 x)^{8} (\frac{1}{2})^{0} : 256 x^{8}

Simplify

\frac{8 !}{1 ! ( 8 - 1 ) !} (- 2 x)^{7} (\frac{1}{2})^{1} : - 512 x^{7}

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} (- 2 x)^{6} (\frac{1}{2})^{2} : 448 x^{6}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} (- 2 x)^{5} (\frac{1}{2})^{3} : - 224 x^{5}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} (- 2 x)^{4} (\frac{1}{2})^{4} : 70 x^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} (- 2 x)^{3} (\frac{1}{2})^{5} : - 14 x^{3}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} (- 2 x)^{2} (\frac{1}{2})^{6} : \frac{7 x ^{2}}{4}

Simplify

\frac{8 !}{7 ! ( 8 - 7 ) !} (- 2 x)^{1} (\frac{1}{2})^{7} : - \frac{x}{8}

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} (- 2 x)^{0} (\frac{1}{2})^{8} : \frac{1}{256}

= 256 x^{8} - 512 x^{7} + 448 x^{6} - 224 x^{5} + 70 x^{4} - 14 x^{3} + \frac{7 x ^{2}}{4} - \frac{x}{8} + \frac{1}{256}

2 x (e^{2 x} + cos (x)) + (e^{2 x} \cdot 2 - sin (x)) x^{2}

\frac{( s ^{2} + 2 s + 2 ) s}{( s + 1 ) ( s ^{2} + 1 )}

(x^{3} + \frac{1}{x ^{3}})^{2}

(e^{2 x} - 1) (e^{2 x} + 1)

(- 12 mn + 5) l e f t (- 3 mn - 2)

What is expand (-2x+1/2)^8 ?
The solution to expand (-2x+1/2)^8 is 256x^8-512x^7+448x^6-224x^5+70x^4-14x^3+(7x^2)/4-x/8+1/256