Wzór dwumianowy Newtona Dla dowolnych liczb rzeczywistych a i b, różnych od zera, oraz dowolnej liczby naturalnej n zachodzi wzór: ${\left(a+b\right)}^n=\left(\begin{array}{c}n\\ 0\end{array}\right)a^n{b}^0+\left(\begin{array}{c}n\\ 1\end{array}\right)a^{n-1}{b}^1+\left(\begin{array}{c}n\\ 2\end{array}\right)a^{n-2}{b}^2+\dots +\left(\begin{array}{c}n\\ n-1\end{array}\right)a^1{b}^{n-1}+\left(\begin{array}{c}n\\ n\end{array}\right)a^0{b}^n$

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a)

Zapisujemy wzór (a-b)² z użyciem symboli Newtona:

${\left(a-b\right)}^2={\left(a+\left(-b\right)\right)}^2=\left(\begin{array}{c}2\\ 0\end{array}\right)a^2{\left(-b\right)}^0+\left(\begin{array}{c}2\\ 1\end{array}\right)a^1{\left(-b\right)}^1+\left(\begin{array}{c}2\\ 2\end{array}\right)a^0{\left(-b\right)}^2=$  

$=\left(\begin{array}{c}2\\ 0\end{array}\right)a^2-\left(\begin{array}{c}2\\ 1\end{array}\right)ab+\left(\begin{array}{c}2\\ 2\end{array}\right)b^2=a^2-2ab+b^2$  

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b)

Zapisujemy wzór (a+b)³ z użyciem symboli Newtona:

${\left(a+b\right)}^3=\left(\begin{array}{c}3\\ 0\end{array}\right)a^3{b}^0+\left(\begin{array}{c}3\\ 1\end{array}\right)a^2{b}^1+\left(\begin{array}{c}3\\ 2\end{array}\right)a^1{b}^2+\left(\begin{array}{c}3\\ 3\end{array}\right)a^0{b}^3=$  

$=\left(\begin{array}{c}3\\ 0\end{array}\right)a^3+\left(\begin{array}{c}3\\ 1\end{array}\right)a^{2}b+\left(\begin{array}{c}3\\ 2\end{array}\right)a{b}^2+\left(\begin{array}{c}3\\ 3\end{array}\right)b^3=a^3+3a^{2}b+3a{b}^2+b^3$