What is expand (2-1/2 x)^5 ?

The solution to expand (2-1/2 x)^5 is 32-40x+20x^2-5x^3+(5x^4)/8-(x^5)/(32)

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expand (2-1/2 x)^5

expand

(2 - \frac{1}{2} x)^{5}

32 - 40 x + 20 x^{2} - 5 x^{3} + \frac{5 x ^{4}}{8} - \frac{x ^{5}}{32}

Solution steps

(2 - \frac{1}{2} x)^{5}

Apply binomial theorem:

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

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

= i = 0 \sum 5 (i 5) \cdot 2^{(5 - i)} (- \frac{1}{2} x)^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 2^{5} (- \frac{1}{2} x)^{0} : 32

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 2^{4} (- \frac{1}{2} x)^{1} : - 40 x

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 2^{3} (- \frac{1}{2} x)^{2} : 20 x^{2}

Simplify

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

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 2^{1} (- \frac{1}{2} x)^{4} : \frac{5 x ^{4}}{8}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 2^{0} (- \frac{1}{2} x)^{5} : - \frac{x ^{5}}{32}

= 32 - 40 x + 20 x^{2} - 5 x^{3} + \frac{5 x ^{4}}{8} - \frac{x ^{5}}{32}

\frac{d}{d x} ((\frac{ln ( x )}{4} - \frac{1}{16}) x^{4}) + c

\frac{1}{4} (- 2 e^{- 2 x} x - e^{- 2 x}) + C

\frac{e ^{- 2 s}}{s ( s + 3 ) ( 2 s + 1 ) ( s + 1 )}

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

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

What is expand (2-1/2 x)^5 ?
The solution to expand (2-1/2 x)^5 is 32-40x+20x^2-5x^3+(5x^4)/8-(x^5)/(32)