Considera os polinómios A, B, C e D

Considera os polinómios A, B, C e D.

\(A = 7{x^2} – 2x + \frac{1}{2}\)

\(B = {x^2} – 4x\)

\(C = 3{x^2} – 4x + \frac{7}{3}\)

\(D = 3{x^2} + \frac{1}{2}x – \frac{2}{3}\)

Determina, apresentando o resultado na forma de um polinómio reduzido e ordenado, e indica o grau desse polinómio:

1. \(A + B\)
   \[\begin{array}{*{20}{l}}{A + B}& = &{\left( {7{x^2} – 2x + \frac{1}{2}} \right) + \left( {{x^2} – 4x} \right)}\\{}& = &{8{x^2} – 6x + \frac{1}{2}}\end{array}\]
   O polinómio é de grau 2.
   
   
2. \(B – C\)
   \[\begin{array}{*{20}{l}}{B – C}& = &{\left( {{x^2} – 4x} \right) – \left( {3{x^2} – 4x + \frac{7}{3}} \right)}\\{}& = &{ – 2{x^2} – \frac{7}{3}}\end{array}\]
   O polinómio é de grau 2.
   
   
3. \(C – D\)
   \[\begin{array}{*{20}{l}}{C – D}& = &{\left( {3{x^2} – \mathop 4\limits_{\left( 2 \right)} x + \frac{7}{3}} \right) – \left( {3{x^2} + \frac{1}{2}x – \frac{2}{3}} \right)}\\{}& = &{ – \frac{9}{2}x + 3}\end{array}\]
   O polinómio é de grau 1.
   
   
4. \(A – \left( {B + C + D} \right)\)
   \[\begin{array}{*{20}{l}}{A – \left( {B + C + D} \right)}& = &{\left( {7{x^2} – 2x + \frac{1}{2}} \right) – \left[ {\left( {{x^2} – 4x} \right) + \left( {3{x^2} – 4x + \frac{7}{3}} \right) + \left( {3{x^2} + \frac{1}{2}x – \frac{2}{3}} \right)} \right]}\\{}& = &{\left( {7{x^2} – 2x + \frac{1}{{\mathop 2\limits_{\left( 3 \right)} }}} \right) – \left( {7{x^2} – \frac{{15}}{2}x + \frac{5}{{\mathop 3\limits_{\left( 2 \right)} }}} \right)}\\{}& = &{\frac{{11}}{2}x – \frac{7}{6}}\end{array}\]
   O polinómio é de grau 1.