Vediamo alcuni esempi di semplificazioni tra monomi con prodotti notevoli e moltiplicazione
{\color{blue}1)} (ab)^2(ab^2)+3(a^3b)(4b^2)-5a(-3ab)^2b=\\
=a^3b^3+12a^3b^3-45a^3b^3=-32a^3b^3\\
{\color{blue}2)}(2a-b)(-2a-b)+(3a-2b)^2-3ab+(2a+b)(a+3b)=\\
{\color{blue}\underline{\color{black}{-4a^2}}}+{\color{green}\underline{\color{black}b^2}}+{\color{blue}\underline{\color{black}9a^2}}+{\color{green}\underline{\color{black}4b^2}}-{\color{yellow}\underline{\color{black}12ab}}-{\color{yellow}\underline{\color{black}3ab}}+{\color{blue}\underline{\color{black}2a^2}}+{\color{yellow}\underline{\color{black}6ab}}+{\color{yellow}\underline{\color{black}ab}}+{\color{green}\underline{\color{black}3b^2}}=\\
\color{black}{\color{blue}\underline{\color{black}{7a^2}}\color{black}+{\color{green}\underline{\color{black}8b^2}}-{\color{yellow}\underline{\color{black}8ab}}}\\
{\color{blue}3)}(2x-y)^2+(x-y)^3+(x-2y)(x+2y)+(x+2y)(2y+3y^2)=\\
{\color{blue}\underline{\color{black}{4x^2}}}+{\color{green}\underline{\color{black}{y^2}}}-{\color{pink}\underline{\color{black}{4xy}}}+{\color{yellow}\underline{\color{black}{x^3}}}-{\color{orange}\underline{\color{black}{y^3}}}-{\color{red}\underline{\color{black}{3x^2y}}}+{\color{purple}\underline{\color{black}{3xy^2}}}+{\color{blue}\underline{\color{black}{x^2}}}-{\color{green}\cancel{\underline{\color{black}{4y^2}}}}+{\color{pink}\underline{\color{black}{2xy}}}+{\color{purple}\underline{\color{black}{3xy^2}}}+{\color{green}\cancel{\underline{\color{black}{4y^2}}}}+{\color{orange}\underline{\color{black}{6y^3}}}=\\
{\color{yellow}\underline{\color{black}{x^3}}}{+\color{blue}\underline{\color{black}{5x^2}}}-{\color{red}\underline{\color{black}{3x^2y}}}-{\color{pink}\underline{\color{black}{2xy}}}+{\color{purple}\underline{\color{black}{6xy^2}}}+{\color{green}\underline{\color{black}{y^2}}}+{\color{orange}\underline{\color{black}{5y^3}}}\\
{\color{blue}4)}(2a+b)(2a-b)(4a^2+b^2)-(3a^2+b^2)^2=\\
(4a^2-b^2)(4a^2+b^2)-(9a^4+b^4+18a^2b^2)=\\
{\color{blue}\underline{\color{black}{16a^4}}}-{\color{green}\underline{\color{black}{b^4}}}-{\color{blue}\underline{\color{black}{9a^4}}}-{\color{blue}\underline{\color{black}{b^4}}}-{\color{yellow}\underline{\color{black}{18a^2b^2}}}=\\
{\color{blue}\underline{\color{black}{7a^4}}}-{\color{blue}\underline{\color{black}{2b^4}}}-{\color{yellow}\underline{\color{black}{18a^2b^2}}}\\
{\color{blue}5)}(a-2)(a-3)(2a+1)+(a-2)^3=\\
(a^2-3a-2a+6)(a-3)+a^3-6a^2+12a-8=\\
(a^2-5a+6)(a-3)+a^3-6a^2+12a-8=\\
{\color{blue}\underline{\color{black}{a^3}}}-{\color{green}\underline{\color{black}{3a^2}}}-{\color{green}\underline{\color{black}{5a^2}}}+{\color{yellow}\underline{\color{black}{15a}}}+{\color{yellow}\underline{\color{black}{6a}}}-18+{\color{blue}\underline{\color{black}{a^3}}}-{\color{green}\underline{\color{black}{6a^2}}}+{\color{yellow}\underline{\color{black}{12a}}}-8=\\
{\color{blue}\underline{\color{black}{2a^2}}}-{\color{green}\underline{\color{black}{14a^2}}}+{\color{yellow}\underline{\color{black}{33a}}}-26