What is expand (x^2(-x/2+4)^4)/(32) ?

The solution to expand (x^2(-x/2+4)^4)/(32) is (x^6}{512}-(x^5)/(16)+\frac{3x^4)/4-4x^3+8x^2

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expand (x^2(-x/2+4)^4)/(32)

expand

\frac{x ^{2} ( - \frac{x}{2} + 4 ) ^{4}}{32}

\frac{x ^{6}}{512} - \frac{x ^{5}}{16} + \frac{3 x ^{4}}{4} - 4 x^{3} + 8 x^{2}

Solution steps

\frac{x ^{2} ( - \frac{x}{2} + 4 ) ^{4}}{32}

(- \frac{x}{2} + 4)^{4} = \frac{x ^{4} - 32 x ^{3} + 384 x ^{2} - 2048 x + 4096}{2 ^{4}}

= \frac{\frac{x ^{4} - 32 x ^{3} + 384 x ^{2} - 2048 x + 4096}{2 ^{4}} x ^{2}}{32}

Multiply

x^{2} \frac{x ^{4} - 32 x ^{3} + 384 x ^{2} - 2048 x + 4096}{2 ^{4}} : \frac{x ^{6} - 32 x ^{5} + 384 x ^{4} - 2048 x ^{3} + 4096 x ^{2}}{16}

= \frac{\frac{x ^{6} - 32 x ^{5} + 384 x ^{4} - 2048 x ^{3} + 4096 x ^{2}}{16}}{32}

Apply the fraction rule:

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

= \frac{x ^{6} - 32 x ^{5} + 384 x ^{4} - 2048 x ^{3} + 4096 x ^{2}}{16 \cdot 32}

Multiply the numbers:

16 \cdot 32 = 512

= \frac{x ^{6} - 32 x ^{5} + 384 x ^{4} - 2048 x ^{3} + 4096 x ^{2}}{512}

Apply the fraction rule:

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

\frac{x ^{6} - 32 x ^{5} + 384 x ^{4} - 2048 x ^{3} + 4096 x ^{2}}{512} = \frac{x ^{6}}{512} - \frac{32 x ^{5}}{512} + \frac{384 x ^{4}}{512} - \frac{2048 x ^{3}}{512} + \frac{4096 x ^{2}}{512}

= \frac{x ^{6}}{512} - \frac{32 x ^{5}}{512} + \frac{384 x ^{4}}{512} - \frac{2048 x ^{3}}{512} + \frac{4096 x ^{2}}{512}

\frac{32 x ^{5}}{512} = \frac{x ^{5}}{16}

\frac{384 x ^{4}}{512} = \frac{3 x ^{4}}{4}

\frac{2048 x ^{3}}{512} = 4 x^{3}

\frac{4096 x ^{2}}{512} = 8 x^{2}

= \frac{x ^{6}}{512} - \frac{x ^{5}}{16} + \frac{3 x ^{4}}{4} - 4 x^{3} + 8 x^{2}

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What is expand (x^2(-x/2+4)^4)/(32) ?
The solution to expand (x^2(-x/2+4)^4)/(32) is (x^6}{512}-(x^5)/(16)+\frac{3x^4)/4-4x^3+8x^2