Determina e indica o grau de cada polinómio obtido
Equações de grau superior ao 1.º: Matematicamente Falando 8 - Parte 2 Pág. 65 Ex. 11
Enunciado
Considera os polinómios:
| $A=7{{x}^{2}}-2x+\frac{1}{2}$ | $B={{x}^{2}}-4x$ | $C=3{{x}^{2}}-4x+\frac{7}{3}$ | $D=3{{x}^{2}}+\frac{1}{2}x-\frac{2}{3}$ |
Determina e indica o grau de cada polinómio obtido:
- $A+B$
- $B-C$
- $C-D$
- $A-(B+C+D)$
Resolução
| $A=7{{x}^{2}}-2x+\frac{1}{2}$ | $B={{x}^{2}}-4x$ | $C=3{{x}^{2}}-4x+\frac{7}{3}$ | $D=3{{x}^{2}}+\frac{1}{2}x-\frac{2}{3}$ |
- Ora,
\[\begin{array}{*{35}{l}}
A+B & = & (7{{x}^{2}}-2x+\frac{1}{2})+({{x}^{2}}-4x) \\
{} & = & 7{{x}^{2}}+{{x}^{2}}-2x-4x+\frac{1}{2} \\
{} & = & 8{{x}^{2}}-6x+\frac{1}{2} \\
\end{array}\]
O polinómio obtido é de grau 2.
- Ora,
\[\begin{array}{*{35}{l}}
B-C & = & ({{x}^{2}}-4x)-(3{{x}^{2}}-4x+\frac{7}{3}) \\
{} & = & {{x}^{2}}-4x-3{{x}^{2}}+4x-\frac{7}{3} \\
{} & = & {{x}^{2}}-3{{x}^{2}}-4x+4x-\frac{7}{3} \\
{} & = & -2{{x}^{2}}-\frac{7}{3} \\
\end{array}\]
O polinómio obtido é de grau 2.
- Ora,
\[\begin{array}{*{35}{l}}
C-D & = & (3{{x}^{2}}-4x+\frac{7}{3})-(3{{x}^{2}}+\frac{1}{2}x-\frac{2}{3}) \\
{} & = & 3{{x}^{2}}-4x+\frac{7}{3}-3{{x}^{2}}-\frac{1}{2}x+\frac{2}{3} \\
{} & = & 3{{x}^{2}}-3{{x}^{2}}-\underset{(2)}{\mathop{4x}}\,-\frac{1}{\underset{(1)}{\mathop{2}}\,}x+\frac{7}{3}+\frac{2}{3} \\
{} & = & -\frac{8}{2}x-\frac{1}{2}x+\frac{9}{3} \\
{} & = & -\frac{9}{2}x+3 \\
\end{array}\]
O polinómio obtido é de grau 1.
- Ora,
\[\begin{array}{*{35}{l}}
A-(B+C+D) & = & (7{{x}^{2}}-2x+\frac{1}{2})-\left[ ({{x}^{2}}-4x)+(3{{x}^{2}}-4x+\frac{7}{3})+(3{{x}^{2}}+\frac{1}{2}x-\frac{2}{3}) \right] \\
{} & = & (7{{x}^{2}}-2x+\frac{1}{2})-({{x}^{2}}-4x+3{{x}^{2}}-4x+\frac{7}{3}+3{{x}^{2}}+\frac{1}{2}x-\frac{2}{3}) \\
{} & = & (7{{x}^{2}}-2x+\frac{1}{2})-({{x}^{2}}+3{{x}^{2}}+3{{x}^{2}}-\underset{(2)}{\mathop{4x}}\,-\underset{(2)}{\mathop{4x}}\,+\frac{1}{\underset{(1)}{\mathop{2}}\,}x+\frac{7}{3}-\frac{2}{3}) \\
{} & = & (7{{x}^{2}}-2x+\frac{1}{2})-(7{{x}^{2}}-\frac{15}{2}x+\frac{5}{3}) \\
{} & = & 7{{x}^{2}}-2x+\frac{1}{2}-7{{x}^{2}}+\frac{15}{2}x-\frac{5}{3} \\
{} & = & 7{{x}^{2}}-7{{x}^{2}}-\underset{(2)}{\mathop{2x}}\,+\frac{15}{\underset{(1)}{\mathop{2}}\,}x+\frac{1}{\underset{(3)}{\mathop{2}}\,}-\frac{5}{\underset{(2)}{\mathop{3}}\,} \\
{} & = & \frac{11}{2}x-\frac{7}{6} \\
\end{array}\]
Alternativa:
Para evitar reduzir os termos semelhantes por duas vezes, será preferível começar por desembaraçar os parêntesis:
\[\begin{array}{*{35}{l}}
A-(B+C+D) & = & (7{{x}^{2}}-2x+\frac{1}{2})-\left[ ({{x}^{2}}-4x)+(3{{x}^{2}}-4x+\frac{7}{3})+(3{{x}^{2}}+\frac{1}{2}x-\frac{2}{3}) \right] \\
{} & = & 7{{x}^{2}}-2x+\frac{1}{2}-{{x}^{2}}+4x-3{{x}^{2}}+4x-\frac{7}{3}-3{{x}^{2}}-\frac{1}{2}x+\frac{2}{3} \\
{} & = & 7{{x}^{2}}-{{x}^{2}}-3{{x}^{2}}-3{{x}^{2}}-2x+4x+4x-\frac{1}{2}x+\frac{1}{2}-\frac{7}{3}+\frac{2}{3} \\
{} & = & \underset{(2)}{\mathop{6x}}\,-\frac{1}{\underset{(1)}{\mathop{2}}\,}x+\frac{1}{\underset{(3)}{\mathop{2}}\,}-\frac{5}{\underset{(2)}{\mathop{3}}\,} \\
{} & = & \frac{11}{2}x-\frac{7}{6} \\
\end{array}\]
O polinómio obtido é de grau 1.
