PRODUCTOS NOTABLES

1.   BINOMIO CUADRADO

·         Suma (a+b)² = a² +2ab + b²

EJ:

(3x + 2y)² = (3x)² + 2(3x) (2y) + (2y)²

                = 9x² +12xy +4y²

·         Resta (a+b)² = a² - 2ab + b²

EJ:

(8m – 5)² = (8m)² – 2 (8m) (5) + (5)²

              = 64m² – 80m + 25

2.   SUMA Y DIFERENCIA DE CUADRADOS (a + b) (a-b) = a² – b²

EJ:

(7x +3y) (7x – 3y) = (7x)² – (3y)²

                           = 49x² – 9y²

3.   BINOMIO AL CUBO

·         Suma (a +b)³ = a³ + 3a²b + 3ab² + b³

EJ:

(2m + 6)³= (2m)³ + 3(2m)²(6) + 3(2m) (6)²+ (6)³

              = 8m³+ 72m² + 216m +216

·         Resta (a - b)³ = a³ - 3a²b + 3ab² - b³

EJ:

(4m – 2n)³= (4m)³ – 3(4m)²(2n) + 3(4m) (2n)² – (2n)³

               = 64m³ – 96m²n + 48mn² – 8n³

4.   TRINOMIO AL CUADRADO

(a +b + c)² = a² +b² + c² + 2ab + 2ac +2bc

EJ:

(2x + 3y – 5z)²= (2x)²+ (3y)²+ (-5z)²+ 2(2x) (3y) + 2(2x) (-5z) + 2(3y) (-5z)

                       = 4x² +9y² + 25z² + 12xy -20xz -30yz

5.   SUMA DE CUBOS

a³ + b³= (a +b) (a² – ab +b²)

EJ:

64m³ + 125n⁶ = (4m + 5n²) {(4m)² – (4m) (5n²) + (5n²)²}

                     =(4m + 5n²) (16m² – 20mn² + 25n⁴)

6.   DIFERENCIA DE CUBOS

a³ – b³ = (a- b) (a²+ab+b²)

EJ:

27x¹² – 8y⁹= (3x⁴ – 2y³) {(3x⁴)² + (3x⁴) (2y³) + (2y³)²}

                = (3x⁴ – 2y³) (9x⁸ + 6x⁴y³+ 4y⁶)

7.   TRIENGULO DE PASCAL

(a + b)⁴= 1a⁴ +4a³b + 6a²b² +4ab³ + 1b⁴

(2x – 3y)⁴= 1(2x)⁴ – 4(2x)³ (2y) + 6(2x)²(3y)² – 4(2x)(3y)³ +1(3y)⁴

               = 16x⁴ – 96x³y +216x²y² -216xy³ + 81y⁴