What is expand (x^2(-x/2+4)^3)/6 ?

The solution to expand (x^2(-x/2+4)^3)/6 is -(x^5}{48}+\frac{x^4)/2-4x^3+(32x^2)/3

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expand (x^2(-x/2+4)^3)/6

expand

\frac{x ^{2} ( - \frac{x}{2} + 4 ) ^{3}}{6}

- \frac{x ^{5}}{48} + \frac{x ^{4}}{2} - 4 x^{3} + \frac{32 x ^{2}}{3}

Solution steps

\frac{x ^{2} ( - \frac{x}{2} + 4 ) ^{3}}{6}

(- \frac{x}{2} + 4)^{3} = \frac{- x ^{3} + 24 x ^{2} - 192 x + 512}{2 ^{3}}

= \frac{\frac{- x ^{3} + 24 x ^{2} - 192 x + 512}{2 ^{3}} x ^{2}}{6}

Multiply

x^{2} \frac{- x ^{3} + 24 x ^{2} - 192 x + 512}{2 ^{3}} : \frac{- x ^{5} + 24 x ^{4} - 192 x ^{3} + 512 x ^{2}}{8}

= \frac{\frac{- x ^{5} + 24 x ^{4} - 192 x ^{3} + 512 x ^{2}}{8}}{6}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{- x ^{5} + 24 x ^{4} - 192 x ^{3} + 512 x ^{2}}{8 \cdot 6}

Multiply the numbers:

8 \cdot 6 = 48

= \frac{- x ^{5} + 24 x ^{4} - 192 x ^{3} + 512 x ^{2}}{48}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- x ^{5} + 24 x ^{4} - 192 x ^{3} + 512 x ^{2}}{48} = - \frac{x ^{5}}{48} + \frac{24 x ^{4}}{48} - \frac{192 x ^{3}}{48} + \frac{512 x ^{2}}{48}

= - \frac{x ^{5}}{48} + \frac{24 x ^{4}}{48} - \frac{192 x ^{3}}{48} + \frac{512 x ^{2}}{48}

Cancel

\frac{24 x ^{4}}{48} : \frac{x ^{4}}{2}

= - \frac{x ^{5}}{48} + \frac{x ^{4}}{2} - \frac{192 x ^{3}}{48} + \frac{512 x ^{2}}{48}

Divide the numbers:

\frac{192}{48} = 4

= - \frac{x ^{5}}{48} + \frac{x ^{4}}{2} - 4 x^{3} + \frac{512 x ^{2}}{48}

Cancel

\frac{512 x ^{2}}{48} : \frac{32 x ^{2}}{3}

= - \frac{x ^{5}}{48} + \frac{x ^{4}}{2} - 4 x^{3} + \frac{32 x ^{2}}{3}

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What is expand (x^2(-x/2+4)^3)/6 ?
The solution to expand (x^2(-x/2+4)^3)/6 is -(x^5}{48}+\frac{x^4)/2-4x^3+(32x^2)/3