Pubblicità

Categoria: Espressioni con monomi e polinomi

Esempi di semplificazioni di espressioni tra monomi:

Scritto il
Pubblicato inAlgebra , Espressioni con monomi e polinomi , Monomi e polinomi

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

Esempi di somma tra monomi (esercizi di base)

Scritto il
Pubblicato inEspressioni con monomi e polinomi , Monomi e polinomi

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}}

Esempi di moltiplicazione tra polinomi

Scritto il
Pubblicato inEspressioni con monomi e polinomi , Monomi e polinomi

La moltiplicazione tra due polinomi ha come risultato un nuovo polinomio, che ha come suoi termini i prodotti parziali, che sono determinati dalla moltiplicazione di ciascun membro del primo polinomio con ciascun membro del secondo. Se vogliamo moltiplicare tra loro più di due polinomi, dobbiamo comunque moltiplicarne prima due, poi moltiplicare il risultato per il fattore successivo e così via, fino ad esauirli tutti.

Vediamo alcuni esempi della moltiplicazione tra due polinomi:

{\color{blue}1)} (x+2y)(2-3z)=2x-3xz+4y-6yz\\
 {\color{blue}2)} (-2m+n^2)(3n+m)=-6mn-2m^2+3n^3+mn^2\\
 {\color{blue}3)} (a-b)(3a^2+ab-b^2)=\\
={\color{blue}\underline{\color{black}3a^3} }+{\color{green}\underline{\color{black}a^2b}}-{\color{purple}\underline{\color{black}ab^2}}-{\color{green}\underline{\color{black}3a^2b}}-{\color{purple}\underline{\color{black}ab^2}}+{\color{red}\underline{\color{black}b^3}}={\color{blue}\underline{\color{black}3a^3}}-{\color{green}\underline{\color{black}2a^2b}}-{\color{purple}\underline{\color{black}2ab^2}}+{\color{red}\underline{\color{black}b^3}}\\
{\color{blue}4)} (2a+b)(a^2-b^2+2ab)=\\
={\color{blue}\underline{\color{black}2a^3}}-{\color{green}\cancel{\underline{\color{black}2ab^2}}}+{\color{purple}\underline{\color{black}4a^2b}}+{\color{purple}\underline{\color{black} a^2b}}-{\color{red}\underline{\color{black}b^3}}+{\color{green}\cancel{\underline{\color{black}2ab^2}}}={\color{blue}\underline{\color{black}2a^3}}+{\color{purple}\underline{\color{black}5a^2b}}-{\color{red}\underline{\color{black}b^3}}\\
{\color{blue}5)}(x^2+2y+3y^2)(2x+2xy)=\\
={\color{blue}\underline{\color{black}2x^3}}+{\color{green}\underline{\color{black}2x^3y}}+{\color{purple}\underline{\color{black}4xy}}+{\color{red}\underline{\color{black}4xy^2}}+{\color{red}\underline{\color{black}6xy^2}}+{\color{pink}\underline{\color{black}6xy^3}}=\\
={\color{blue}\underline{\color{black}2x^3}}+{\color{green}\underline{\color{black}2x^3y}}+{\color{purple}\underline{\color{black}4xy}}+{\color{red}\underline{\color{black}10xy^2}}+{\color{pink}\underline{\color{black}6xy^3}}\\
 {\color{blue}6)}\Big(\frac13x-\frac25y\Big)(2x^2+3xy-y^2)=\\
={\color{blue}\underline{\color{black}\frac23x^3}}+{\color{green}\underline{\color{black}x^2y}}-{\color{purple}\underline{\color{black}\frac13xy^2}}-{\color{green}\underline{\color{black}\frac45x^2y}}-{\color{purple}\underline{\color{black}\frac65xy^2}}+{\color{red}\underline{\color{black}\frac25y^3}}=\\
={\color{blue}\underline{\color{black}\frac23x^3}}+{\color{green}\underline{\color{black}\frac15x^2y}}-{\color{purple}\underline{\color{black}\frac{23}{15}xy^2}}+{\color{red}\underline{\color{black}\frac25y^3}}\\
{\color{blue}7)}(x+2xy-y)(y-4x^2+2x^2y)=\\
{\color{blue}\underline{\color{black}xy}}-{\color{green}\underline{\color{black}4x^3}}+{\color{purple}\underline{\color{black}2x^3y}}+{\color{pink}\underline{\color{black}2xy^2}}-{\color{purple}\underline{\color{black}8x^3y}}+{\color{yellow}\underline{\color{black}4x^3y^2}}-{\color{orange}\underline{\color{black}y^2}}+{\color{skyblue}\underline{\color{black}4x^2y}}-{\color{red}\underline{\color{black}2x^2y^2}}=\\
={\color{yellow}\underline{\color{black}4x^3y^2}}-{\color{purple}\underline{\color{black}6x^3y}}-{\color{green}\underline{\color{black}4x^3}}-{\color{red}\underline{\color{black}2x^2y^2}}+{\color{skyblue}\underline{\color{black}4x^2y}}+{\color{pink}\underline{\color{black}2xy^2}}+{\color{blue}\underline{\color{black}xy}}-{\color{orange}\underline{\color{black}y^2}}

Esempi di somma tra monomi (esercizi di base)

Scritto il
Pubblicato inEspressioni con monomi e polinomi , Monomi e polinomi

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}}