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Popolare Algebra >

espandere (x+δx)^5

Soluzione

espandere

(x + δ x)^{5}

Soluzione

x^{5} + 5 δ x^{5} + 10 δ^{2} x^{5} + 10 δ^{3} x^{5} + 5 δ^{4} x^{5} + δ^{5} x^{5}

Fasi della soluzione

(x + δ x)^{5}

Applicare il teorema binomiale:

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

a = x, b = δ x

= i = 0 \sum 5 (i 5) x^{(5 - i)} (δ x)^{i}

Espandere la sommatoria

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

Semplifica

\frac{5 !}{0 ! ( 5 - 0 ) !} x^{5} (δ x)^{0} : x^{5}

Semplifica

\frac{5 !}{1 ! ( 5 - 1 ) !} x^{4} (δ x)^{1} : 5 δ x^{5}

Semplifica

\frac{5 !}{2 ! ( 5 - 2 ) !} x^{3} (δ x)^{2} : 10 δ^{2} x^{5}

Semplifica

\frac{5 !}{3 ! ( 5 - 3 ) !} x^{2} (δ x)^{3} : 10 δ^{3} x^{5}

Semplifica

\frac{5 !}{4 ! ( 5 - 4 ) !} x^{1} (δ x)^{4} : 5 δ^{4} x^{5}

Semplifica

\frac{5 !}{5 ! ( 5 - 5 ) !} x^{0} (δ x)^{5} : δ^{5} x^{5}

= x^{5} + 5 δ x^{5} + 10 δ^{2} x^{5} + 10 δ^{3} x^{5} + 5 δ^{4} x^{5} + δ^{5} x^{5}

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