Calcula
Monómios e polinómios: Matematicamente Falando 8 - Pág. 136 Ex. 1
Enunciado
Calcula:
| \({\left( {2x – 3} \right)^2}\) | \({\left( {x + 7} \right)^2}\) | \({\left( {y + \frac{1}{2}} \right)^2}\) |
| \({\left( {4a – 3b} \right)^2}\) | \({\left( { – x – 1} \right)^2}\) | \({\left( {x + 1} \right)^2}\) |
Resolução
Quadrado do binómio
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
O quadrado de um binómio obtém-se adicionando o quadrado do primeiro monómio ao dobro do produto do primeiro monómio pelo segundo monómio e ao quadrado do segundo monómio.
| a) | \(\begin{array}{*{20}{l}}{{{\left( {2x – 3} \right)}^2}}& = &{{{\left( {2x} \right)}^2} + 2 \times \left( {2x} \right) \times \left( { – 3} \right) + {{\left( { – 3} \right)}^2}}\\{}& = &{4{x^2} – 12x + 9}\end{array}\) |
| b) | \(\begin{array}{*{20}{l}}{{{\left( {x + 7} \right)}^2}}& = &{{x^2} + 2 \times x \times 7 + {7^2}}\\{}& = &{{x^2} + 14x + 49}\end{array}\) |
| c) | \(\begin{array}{*{20}{l}}{{{\left( {y + \frac{1}{2}} \right)}^2}}& = &{{y^2} + 2 \times y \times \frac{1}{2} + {{\left( {\frac{1}{2}} \right)}^2}}\\{}& = &{{y^2} + y + \frac{1}{4}}\end{array}\) |
| d) | \(\begin{array}{*{20}{l}}{{{\left( {4a – 3b} \right)}^2}}& = &{{{\left( {4a} \right)}^2} + 2 \times 4a \times \left( { – 3b} \right) + {{\left( { – 3b} \right)}^2}}\\{}& = &{16{a^2} – 24ab + 9{b^2}}\end{array}\) |
| e) | \(\begin{array}{*{20}{l}}{{{\left( { – x – 1} \right)}^2}}& = &{{{\left( { – x} \right)}^2} + 2 \times \left( { – x} \right) \times \left( { – 1} \right) + {{\left( { – 1} \right)}^2}}\\{}& = &{{x^2} + 2x + 1}\end{array}\) |
| f) | \(\begin{array}{*{20}{l}}{{{\left( {x + 1} \right)}^2}}& = &{{x^2} + 2 \times x \times 1 + {1^2}}\\{}& = &{{x^2} + 2x + 1}\end{array}\) |
