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Popular Álgebra >

expandir (2x-1/2 y)^5

Solução

expandir

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

Solução

\frac{1024 x ^{5} - y ^{5}}{32} - 40 x^{4} y + 20 x^{3} y^{2} - 5 x^{2} y^{3} + \frac{5 x y ^{4}}{8}

Passos da solução

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

Aplicar o teorema do binômio:

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

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

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

Expandir a soma

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

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

Fatorar

32 x^{5} - 40 x^{4} y + 20 x^{3} y^{2} - 5 x^{2} y^{3} + \frac{5 x y ^{4}}{8} - \frac{y ^{5}}{32} : \frac{1024 x ^{5} - y ^{5}}{32} - 40 x^{4} y + 20 x^{3} y^{2} - 5 x^{2} y^{3} + \frac{5 x y ^{4}}{8}

= \frac{1024 x ^{5} - y ^{5}}{32} - 40 x^{4} y + 20 x^{3} y^{2} - 5 x^{2} y^{3} + \frac{5 x y ^{4}}{8}

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expandir T2^2(T^2-1.9044=2405.6014)

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