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# Mathematrix: Aufgabenbeispiele/ Das pascalsche Dreieck Binompotenzen

Mit Hilfe des Pascalschen Dreiecks multiplizieren Sie die Klammern aus:

${\displaystyle \quad }$A)${\displaystyle \ \left(a^{3}-4\right)^{5}\quad }$ ${\displaystyle \quad }$B)${\displaystyle \ \left(5\ x^{2}+4\ z^{5}\right)^{4}\quad }$

${\displaystyle \quad }$A)${\displaystyle \ \left(a^{3}-4\right)^{5}=}$
${\displaystyle \quad }$${\displaystyle {\color {red}\mathbf {1} }\cdot \left(a^{3}\right)^{5}\ {\cancelto {1}{\ 4^{0}\ }}-{\color {red}\mathbf {5} }\cdot \left(a^{3}\right)^{4}\ 4^{1}+{\color {red}\mathbf {1} 0}\cdot \left(a^{3}\right)^{3}\ 4^{2}-{\color {red}\mathbf {1} 0}\cdot \left(a^{3}\right)^{2}\ 4^{3}+{\color {red}\mathbf {5} }\cdot \left(a^{3}\right)^{1}\ 4^{4}-{\color {red}\mathbf {1} }\cdot {\cancelto {1}{\ \left(a^{3}\right)^{0}\ }}\ 4^{5}}$
${\displaystyle \quad }$${\displaystyle a^{15}\ -20\cdot a^{12}\ +160\cdot a^{9}\ -640\cdot a^{6}\ +1280\cdot a^{3}\ -1024\ }$

${\displaystyle \quad }$B)${\displaystyle \ \left(5\ x^{2}+4\ z^{5}\right)^{4}=\quad }$
${\displaystyle \quad }$${\displaystyle {\color {red}\mathbf {1} }\cdot \left(5\ x^{2}\right)^{4}\ {\cancelto {1}{\ \left(4\ z^{5}\right)^{0}\ }}+{\color {red}\mathbf {4} }\cdot \left(5\ x^{2}\right)^{3}\ \left(4\ z^{5}\right)^{1}+{\color {red}\mathbf {6} }\cdot \left(5\ x^{2}\right)^{2}\ \left(4\ z^{5}\right)^{2}+{\color {red}\mathbf {4} }\cdot \left(5\ x^{2}\right)^{1}\ \left(4\ z^{5}\right)^{3}+{\color {red}\mathbf {1} }\cdot {\cancelto {1}{\ \left(5\ x^{2}\right)^{0}\ }}\ \left(4\ z^{5}\right)^{4}}$
${\displaystyle \quad }$${\displaystyle 625\cdot x^{8}\ z^{5}\ +2000\cdot x^{6}\ z^{10}\ +2400\cdot x^{4}\ z^{15}\ 1280\cdot x^{2}\ z^{20}\ +256\cdot z^{25}\ \ }$