Ricordiamo che un monomio è un’espressione algebrica, costituita soltanto da prodotti di fattori, sia numerici che letterali. Due monomi si dicono simili se hanno stessa parte letterale e in tal caso possiamo sommarli (sommandone i coefficienti), altrimenti non è possibile.
Vediamo alcuni esempi:
{\color{blue}1)}-{\color{blue}\underline{\color{black}a}}+{\color{green}\underline{\color{black}3b}}+{\color{purple}\underline{\color{black}2c}}+{\color{blue}\underline{\color{black}a}}+{\color{green}\underline{\color{black}2b}}=\\
={\color{blue}\underline{\color{black}(-1+1)\cdot a}}+{\color{green}\underline{\color{black}(3+2)\cdot b}}+{\color{purple}\underline{\color{black}2c}}={\color{blue}\underline{\color{black}0\cdot a}}+{\color{green}\underline{\color{black}5\cdot b}}+{\color{purple}\underline{\color{black}2c}}=\\
={\color{green}\underline{\color{black}5b}}+{\color{purple}\underline{\color{black}2c}}\\
{\color{blue}2)} {\color{blue}\underline{\color{black}5x}}+{\color{pink}\underline{\color{black}2y}}-{\color{red}\underline{\color{black}z}}+{\color{red}\underline{\color{black}3z}}-{\color{blue}\underline{\color{black}x}}=\\
={\color{blue}\underline{\color{black}(5-1)\cdot x}}+{\color{pink}\underline{\color{black}2y}}+{\color{red}\underline{\color{black}(-1+3)\cdot z}}={\color{blue}\underline{\color{black}4x}}+{\color{pink}\underline{\color{black}2y}}+{\color{red}\underline{\color{black}2z}}\\
{\color{blue}3)}-{\color{blue}\underline{\color{black}8x^3y}}+{\color{green}\underline{\color{black}3x^2y}}+{\color{purple}\underline{\color{black}2xy}}+{\color{blue}\underline{\color{black}x^3y}}-{\color{purple}\underline{\color{black}4xy}}+{\color{blue}\underline{\color{black}5x^3y}}=\\
={\color{blue}\underline{\color{black}(-8+1+5)\cdot x^3y}}+{\color{green}\underline{\color{black}3x^2y}}+{\color{purple}\underline{\color{black}(2-4)\cdot xy}}=-{\color{blue}\underline{\color{black}2x^3y}}+{\color{green}\underline{\color{black}3x^2y}}-{\color{purple}\underline{\color{black}2xy}}\\
{\color{blue}4)}{\color{blue}\underline{\color{black}ax}}+{\color{grren}\underline{\color{black}7ab}}+{\color{purple}\underline{\color{black}7ac}}+{\color{pink}\underline{\color{black}x}}+{\color{purple}\underline{\color{black}ac}}-{\color{blue}\underline{\color{black}6ax}}+{\color{green}\underline{\color{black}8ab}}+{\color{blue}\underline{\color{black}5ax}}=\\
={\color{blue}\underline{\color{black}(1-6+5)\cdot ax}}+{\color{green}\underline{\color{black}(7+8)\cdot ab}}+{\color{purple}\underline{\color{black}(7+1)\cdot ac}}+{\color{pink}\underline{\color{black}x}}=\\
={\color{blue}\underline{\color{black}0\cdot ax}}+{\color{green}\underline {\color{black}15\cdot ab}}+{\color{purple}\underline{\color{black}8\cdot ac}}+{\color{pin}\underline{\color{black}x}}=\\
={\color{green}\underline{\color{black}15ab}}+{\color{purple}\underline{\color{black}8ac}}+{\color{pink}\underline{\color{black}x}}\\
{\color{blue}5)}{\color{blue}\underline{\color{black}3x}}+{\color{green}\underline{\color{black}6x^2}}+{\color{red}\underline{\color{black}2}}+{\color{purple}\underline{\color{black}x^4}}-{\color{blue}\underline{\color{black}6x}}-{\color{red}\underline{\color{black}7}}+{\color{green}\underline{\color{black}7x^2}}-{\color{pink}\underline{\color{black}6x^3}}=\\
={\color{purple}\underline{\color{black}x^4}}-{\color{pink}\underline{\color{black}x^3}}+{\color{green}\underline{\color{black}(6+7)\cdot x^2}}+{\color{blue}\underline{\color{black}(3-6)\cdot x}}+{\color{red}\underline{\color{black}(2-7)}}=\\
={\color{purple}\underline{\color{black}x^4}}-{\color{pink}\underline{\color{black}x^3}}+{\color{green}\underline{\color{black}13x^3}}-{\color{blue}\underline{\color{black}3x}}-{\color{red}\underline{\color{black}5}}\\
{\color{blue}6)}{\color{blue}\underline{\color{black}6x^2}}+{\color{green}\underline{\color{black}xy}}-{\color{purple}\underline{\color{black}y^3}}+{\color{pink}\underline{\color{black}y}}-{\color{green}\underline{\color{black}2xy}}-{\color{blue}\underline{\color{black}x^2}}+{\color{yellow}\underline{\color{black}2y^2}}=\\
={\color{blue}\underline{\color{black}(6-1)\cdot x^2}}+{\color{green}\underline{\color{black}(1-2)\cdot xy}}+{\color{pink}\underline{\color{black}y}}+{\color{yellow}\underline{\color{black}2y^2}}-{\color{purple}\underline{\color{black}y^3}}=\\
={\color{blue}\underline{\color{black}5x^2}}-{\color{green}\underline{\color{black}xy}}+{\color{pink}\underline{\color{black}y}}+{\color{yellow}\underline{\color{black}2y^2}}-{\color{purple}\underline{\color{black}y^3}}\\Ora vediamo altri esempi, nei quali introduciamo le parentesi (ricordiamo che se abbiamo un + davanti, si mantengono gli stessi segni, se c’è un – si invertono):
- Togliamo le parentesi
- Procediamo come nei casi precedenti
{\color{blue}1)}(-ab+4ab^2+c)+(-b-3a^2b)-(-c-ab^2+2a^2b)=\\
=-{\color{blue}\underline{\color{black}ab}}+{\color{green}\underline{\color{black}4ab^2}}+{\color{purple}\underline{\color{black}c}}-{\color{red}\underline{\color{black}b}}-{\color{pink}\underline{\color{black}3a^2b}}+{\color{purple}\underline{\color{black}c}}+{\color{green}\underline{\color{black}ab^2}}-{\color{pink}\underline{\color{black}2a^2b}}=\\
={\color{pink}\underline{\color{black}(-3-2)\cdot a^2b}}+{\color{green}\underline{\color{black}(4+1)\cdot ab^2}}-{\color{blue}\underline{\color{black}ab}}+{\color{red}\underline{\color{black}b}}+{\color{purple}\underline{\color{black}(1+1)\cdot c}}=\\
={\color{pink}\underline{\color{black}-5a^2b}}+{\color{green}\underline{\color{black}5ab^2}}-{\color{blue}\underline{\color{black}ab}}+{\color{red}\underline{\color{black}b}}+{\color{purple}\underline{\color{black}2c}}\\
{\color{blue}2)}(5xy^2-3x^2y+xy)-(-x^2y+4xy)+6x^2y-(-7xy+6x^2y)=\\
={\color{blue}\underline{\color{black}5xy^2}}-{\color{green}\underline{\color{black}3x^2y}}+{\color{red}\underline{\color{black}xy}}+{\color{green}\underline{\color{black}x^2y}}-{\color{red}\underline{\color{black}4xy}}+{\color{green}\cancel{\underline{\color{black}6x^2y}}}+{\color{red}\underline{\color{black}7xy}}-{\color{green}\cancel{\underline{\color{black}6x^2y}}}=\\
={\color{green}\underline{\color{black}(-3+1)\cdot x^2y}}+{\color{blue}\underline{\color{black}5x^2y}}+{\color{red}\underline{\color{black}(1-4+7)\cdot xy}}=\\
={\color{green}\underline{\color{black}-2x^2y}}+{\color{blue}\underline{\color{black}5x^2y}}+{\color{red}\underline{\color{black}4xy}}\\
{\color{blue}3)}\Big(\frac34ab^2+\frac12x^2\Big)-\Big(x^2-\frac13ab^2\Big)=\\
={\color{blue}\underline{\color{black}\frac34ab^2}}+{\color{green}\underline{\color{black}\frac12x^2}}-{\color{green}\underline{\color{black}x^2}}+{\color{blue}\underline{\color{black}\frac13ab^2}}=\\
={\color{blue}\underline{\color{black}\Big(\frac34+\frac13\Big)\cdot ab^2}}+{\color{gree}\underline{\color{black}\Big(\frac12-1\Big)\cdot x^2}}={\color{blue}\underline{\color{black}\frac{13}{12}ab^2}}-{\color{blue}\underline{\color{black}\frac12x^2}}