Decompõe em fatores os polinómios
Equações de grau superior ao 1.º: Matematicamente Falando 8 - Parte 2 Pág. 75 Ex. 20
Enunciado
Decompõe em fatores os polinómios:
- ${{x}^{2}}-6x+9$
- $4{{x}^{2}}+4x+1$
- ${{a}^{2}}+2ab+{{b}^{2}}$
- ${{y}^{2}}-25$
- $4{{a}^{2}}-1$
- $8{{x}^{3}}y-2x{{y}^{3}}$
- $2{{x}^{2}}+12x+18$
- $3{{a}^{2}}x+6ax+3x$
- ${{x}^{3}}-x$
- ${{a}^{2}}(a-2)-2a(a-2)+(a-2)$
Resolução
- Ora,
$\begin{array}{*{35}{l}}
{{x}^{2}}-6x+9 & = & {{(x-3)}^{2}} \\
{} & = & (x-3)(x-3) \\
\end{array}$ - Ora,
$\begin{array}{*{35}{l}}
4{{x}^{2}}+4x+1 & = & {{(2x+1)}^{2}} \\
{} & = & (2x+1)(2x+1) \\
\end{array}$ - Ora,
$\begin{array}{*{35}{l}}
{{a}^{2}}+2ab+{{b}^{2}} & = & {{(a+b)}^{2}} \\
{} & = & (a+b)(a+b) \\
\end{array}$ - Ora,
${{y}^{2}}-25=(y+5)(y-5)$ - Ora,
$4{{a}^{2}}-1=(2a+1)(2a-1)$ - Ora,
$\begin{array}{*{35}{l}}
8{{x}^{3}}y-2x{{y}^{3}} & = & 2xy(4{{x}^{2}}-{{y}^{2}}) \\
{} & = & 2xy(2x+y)(2x-y) \\
\end{array}$ - Ora,
$\begin{array}{*{35}{l}}
2{{x}^{2}}+12x+18 & = & 2({{x}^{2}}+6x+9) \\
{} & = & 2{{(x+3)}^{2}} \\
{} & = & 2(x+3)(x+3) \\
\end{array}$ - Ora,
$\begin{array}{*{35}{l}}
3{{a}^{2}}x+6ax+3x & = & 3x({{a}^{2}}+2a+1) \\
{} & = & 3x{{(a+1)}^{2}} \\
{} & = & 3x(a+1)(a+1) \\
\end{array}$ - Ora,
$\begin{array}{*{35}{l}}
{{x}^{3}}-x & = & x({{x}^{2}}-1) \\
{} & = & x(x+1)(x-1) \\
\end{array}$ - Ora,
$\begin{array}{*{35}{l}}
{{a}^{2}}(a-2)-2a(a-2)+(a-2) & = & (a-2)({{a}^{2}}-2a+1) \\
{} & = & (a-2){{(a-1)}^{2}} \\
{} & = & (a-2)(a-1)(a-1) \\
\end{array}$
![Um quadrado [ABCD]](https://www.acasinhadamatematica.pt/wp-content/uploads/2018/04/9V2Pag92-1a-720x340.png)
