espandere (3x^3-4y^2)^5

Soluzione

espandere

(3 x^{3} - 4 y^{2})^{5}

243 x^{15} - 1620 x^{12} y^{2} + 4320 x^{9} y^{4} - 5760 x^{6} y^{6} + 3840 x^{3} y^{8} - 1024 y^{10}

Fasi della soluzione

(3 x^{3} - 4 y^{2})^{5}

Applicare il teorema binomiale:

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

a = 3 x^{3}, b = - 4 y^{2}

= i = 0 \sum 5 (i 5) (3 x^{3})^{(5 - i)} (- 4 y^{2})^{i}

Espandere la sommatoria

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

Semplificare

\frac{5 !}{0 ! ( 5 - 0 ) !} (3 x^{3})^{5} (- 4 y^{2})^{0} : 243 x^{15}

Semplificare

\frac{5 !}{1 ! ( 5 - 1 ) !} (3 x^{3})^{4} (- 4 y^{2})^{1} : - 1620 x^{12} y^{2}

Semplificare

\frac{5 !}{2 ! ( 5 - 2 ) !} (3 x^{3})^{3} (- 4 y^{2})^{2} : 4320 x^{9} y^{4}

Semplificare

\frac{5 !}{3 ! ( 5 - 3 ) !} (3 x^{3})^{2} (- 4 y^{2})^{3} : - 5760 x^{6} y^{6}

Semplificare

\frac{5 !}{4 ! ( 5 - 4 ) !} (3 x^{3})^{1} (- 4 y^{2})^{4} : 3840 x^{3} y^{8}

Semplificare

\frac{5 !}{5 ! ( 5 - 5 ) !} (3 x^{3})^{0} (- 4 y^{2})^{5} : - 1024 y^{10}

= 243 x^{15} - 1620 x^{12} y^{2} + 4320 x^{9} y^{4} - 5760 x^{6} y^{6} + 3840 x^{3} y^{8} - 1024 y^{10}

s im pl i f y \frac{20}{75}

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s im pl i f y 18 1^{2} - 1 9^{2}

s im pl i f y - 8 \cdot 1

s im pl i f y - 3 (- 5)