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PRODUCTOS NOTABLES Y FACTORIZACIÓN

Los productos notables son aquellos productos de expresiones algebraicas que se pueden resolver con la ayuda de reglas generales y evitar que se hagan todas las operaciones de desarrollo.

Los productos notables más comunes son:

          1.-Binomio al cuadrado (x+ y)2

          2.-Binomios conjugados (x + y) (x – y)

          3.-Binomios con termino común (x + a) (x + b)

          4.-Binomia al cubo (x + b)3

BINOMIO AL CUADRADO: (x+ y)2


RESULTADO:

Trinomio al cuadrado perfecto      

REGLAS PARA DESARROLLAR EL BINOMIO AL CUADRADO:

1.-Se eleva al cuadrado el primer término del binomio

2.-Se suma o se resta el doble producto del primer término por el segundo

3.-Se suma el cuadrado del segundo término del binomio

Ejemplos de binomio al cuadrado:

BINOMIOS CONJUGADOS (x + y) (x – y):


RESULTADO:

Diferencia de cuadrados (x2 – y2)

REGLAS PARA DESARROLLAR LOS BINOMIOS CONJUGADOS:

1.-Se eleva al cuadrado el término que no cambia de signo

2.-Se resta el cuadrado del término que cambia de signo

Ejemplos de binomios conjugados:

BINOMIOS CON TERMINO COMÚN (x + a) (x + b)


RESULTADO:

Trinomio de la forma:  x2 + (a + b)x + ab

REGLAS PARA DESARROLLAR BINOMIOS CON TÉRMINO COMÚN:

1.-Se eleva al cuadrado el termino común

2.-Se suman algebraicamente los términos no comunes y se multiplican por el término común

3.-Se suma el producto algebraico de los dos términos no comunes

Ejemplos de binomios con término común:

BINOMIO AL CUBO (x + b)3


RESULTADO:

Polinomio de la forma:  x3 + 3x2y + 3xy2 + y3

REGLAS PARA DESARROLLAR UN TRINOMIO AL CUBO:

1.-El cubo el primer término

2.-Mas el triple producto del cuadrado del primer término por el segundo

3.-Mas el triple producto del primer término por el cuadrado del segundo termino

4.-Mas el cubo del segundo término

Ejemplos de binomio al cubo:

FACTORIZACIÓN

La factorización es el proceso algebraico por medio del cual se transforma una suma o resta de términos algebraicos en un producto algebraico.

También se puede entender como el proceso inverso del desarrollo de productos notables.


FACTOR COMÚN


REGLAS PARA OBTENER EL FACTOR COMÚN DE UN POLINOMIO

1.- Se obtiene el máximo común divisor de los coeficientes

2.-Se identifica las literales con menor exponente que se repitan en cada uno de los términos algebraicos del polinomio a factorizar.

Ejemplos de factor común en un polinomio:

FACTORIZACIÓN POR AGRUPACIÓN


En la factorización por agrupación, no todos los elementos del polinomio comparten un factor común, por lo que se deben identificar primero los grupos de elementos que si comparten términos comunes y después factorizar cada grupo de elementos.

Ejemplos de factorización por agrupación:

DIFERENCIA DE CUADRADOS


La diferencia de cuadrados tiene la forma de x2 – y2   y su factorización es el producto de binomios conjugados:

Ejemplos de diferencia de cuadrados:

TRINOMIO AL CUADRADO PERFECTO


RESULTADO:

Es un binomio al cuadrado

REGLAS PARA FACTORIZAR UN TRINOMIO AL CUADRADO PERFECTO

1.-Se ordenan los términos del trinomio en orden descendente a una de las literales, de forma que los extremos sean expresiones que tengan raíz cuadrada exacta

2.-Se obtiene la raíz del primer y tercer termino

3.-Para comprobar que haya sido un trinomio al cuadrado “perfecto”, se realiza el doble producto de los términos obtenidos en el paso dos y debe ser igual al 2do término del trinomio.

4.-El signo del binomio que dio resultado es el mismo que el signo del 2do término del trinomio original

Ejemplos de Trinomio al Cuadrado Perfecto:

TRINOMIO DE LA FORMA: (x2 + bx + c)


RESULTADO:

Producto de dos binomios con término común

REGLAS PARA FACTORIZAR UN TRINOMIO DE LA FORMA (x2 + bx + c):

1.-Se ordenan los términos del trinomio en orden descendente a una de las literales, de forma que el primer término sea una expresión que tenga raíz cuadrada exacta

2.-Se obtiene la raíz cuadrada de este primer término y se coloca en los dos binomios

3.-Se buscan dos números que su producto sea igual al 3er término del trinomio (c) y su suma aritmética sea igual al coeficiente del 2do término del trinomio (b). De estos números, el mayor se coloca en el primer binomio y el menor en el segundo binomio.

x2 + (e + h) x + (e * h) = (x + e) (x + h)

4.-El signo del primer binomio es igual al signo del 2do término del trinomio, y el signo del segundo binomio es igual al signo resultante del producto de los signos del 2do por el 3er término del trinomio.

Ejemplos de trinomios de la forma (x2 + bx + c):

TRINOMIO DE LA FORMA (ax2 + bx + c)


RESULTADO:

Producto de dos binomios

REGLAS PARA FACTORIZAR UNA TRINOMIO DE LA FORMA (ax2 + bx + c):

1.-Se ordenan los términos del trinomio en orden descendente a una de las literales

2.-Se multiplica y se divide el trinomio por el coeficiente del 1er termino

3.-Con esto, el trinomio del numerador se factoriza a dos binomios con termino común

4.-A cada uno de estos dos binomios se les divide por el denominador para obtener los dos binomios de la forma: (fx + e) (x + h)

Ejemplos de trinomios de la forma (ax2 + bx + c):

SUMA Y DIFERENCIA DE CUBOS


RESULTADO:

Producto de un binomio por un trinomio

REGLAS PARA FACTORIZAR UNA SUMA O DIFERENCIA DE CUBOS:

1.-Se obtienen las raíces de cada uno de los términos

2.-El primer término es un binomio igual a la suma o resta de estas raíces obtenidas

3.-El segundo término es un trinomio igual:

          1er termino: Igual al cuadrado de la raíz del primer término del binomio

          2do término: Igual al producto de las raíces del binomio con signo opuesto.

          3er término: Igual al cuadrado de la raíz del segundo término del binomio

Ejemplos de Suma y diferencia de cubos:

Mas ejemplos, ejercicios y preguntas clave de examen, en nuestras guías digitales.

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Curso para la UNAM

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