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Prodotti notevoli

Somma per differenza

\[\left ( A+B \right )\left ( A-B \right )=A^{2}-B^{2}\]

Esempio 1.- Calcolare i seguenti  prodotti \[\left ( 3x+2y \right )\left ( 3x-2y \right ),\, \left ( 4x-5y \right )\left ( 4x+5y \right )\]

Calcolo

\[\left ( 3x+2y \right )\left ( 3x-2y \right )=\left ( 3x \right )^{2}-\left ( 2y \right )^{2}=9x^{2}-4y^{2}\]

\[\left ( 4x-5y \right )\left ( 4x+5y \right )=16x^{2}-25y^{2}\]

Quadrato del binomio

\[\left ( A\pm B \right )^{2}=A^{2}\pm 2AB+B^{2}\]

Esempio 1.- Calcolare i seguenti quadratidi un binomio \[\left ( 3x+4y \right )^{2},\left ( 5x-6y \right )^{2},\left ( x^{2}+y \right )^{2},\left ( \frac{1}{2}x-\frac{1}{3}y \right )^{2}\]

Calcolo

\[\left ( 3x+4y \right )^{2}=\left ( 3x \right )^{2}+2\left ( 3x \right )\left ( 4y \right )+\left ( 4y \right )^{2}=9x^{2}+24xy+16y^{2}\]

\[\left ( 5x-4y \right )^{2}=\left ( 5x \right )^{2}-2\left ( 5x \right )\left ( 4y \right )+\left ( 4y \right )^{2}=25x^{2}-40xy+16y^{2}\]

\[\left ( \frac{1}{2}x-\frac{1}{3}y \right )^{2}=\left ( \frac{1}{2}x \right )^{2}-2\left ( \frac{1}{2}x \right )\left ( \frac{1}{3}y \right )+\left ( \frac{1}{3}y \right )^{2}=\frac{1}{4}x^{2}-\frac{1}{3}xy+\frac{1}{9}y^{2}\]

Cubo del Binomio

\[\left ( A+ B \right )^{3}=A^{3}+ 3A^{2}B+3AB^{2}+B^{3}\]

\[\left ( A-B \right )^{3}=A^{3}- 3A^{2}B+3AB^{2}-B^{3}\]

Esempio 1.- Calcolare i seguenti cubi di un binomio \[\left ( -x+2y \right )^{3},\left ( 3x-y \right )^{3},\left ( x^{2}+3y \right )^{3},\left ( \frac{1}{2}x-\frac{1}{3}y \right )^{3}\]

Calcolo

\[\left ( -x+2y \right )^{3}=\left ( -x \right )^{3}+3\left ( -x \right )^{2}\left ( 2y \right )+3\left ( -x \right )\left ( 2y \right )^{2}+\left ( 2y \right )^{3}=-x^{3}+3\left ( +x \right )^{2}\left ( 2y \right )+3\left ( -x \right )\left ( 4y^{2} \right )+8y^{3}=-x^{3}+6x^{2}y-12xy^{2}+8y^{3}\]

\[\left ( x^{2}+3y \right )^{3}=\left ( x^{2} \right )^{3}+3\left ( x^{2} \right )^{2}\left ( 3y \right )+3\left ( x^{2} \right )\left ( 3y \right )^{2}+\left ( 3y \right )^{3}=x^{6}+3x^{4}\left ( 3y \right )+3x^{2}\left ( 9y^{2} \right )+27y^{3}=x^{6}+9x^{4}y+27x^{2}y^{2}+27y^{3}\]

Quadrato del trinomio

\[\left ( A+B+C \right )^{2}=A^{2}+B^{2}+C^{2}+2AB+2AC+2BC\]

Esempio 1.- Calcolare i seguenti quadrati di un trinomio \[\left ( 3x+2y+5 \right )^{2},\left ( -x+4y-3z \right )^{2},\left ( \frac{1}{2}+2y+3x \right )^{2},\left ( \frac{2}{3}ab^{2}-ac^{2}+\frac{3}{2}b^{3} \right )^{2},\]

Calcolo

\[\left ( -x+4y-3z \right )^{2}=\left ( -x \right )^{2}+\left ( 4y \right )^{2}+\left ( -3z \right )^{2}+2\left ( -x \right )\left ( 4y \right )+2\left ( -x \right )\left ( -3z \right )+2\left ( 4y \right )\left ( -3z \right )=x^{2}+16y^{2}+9z^{2}-8xy+6xz-24xz\]