Ejercicios de factorizacion resueltos

FACTORIZACION
1.- FACTOR COMÚN
2xy+4y+2yz=2y(x+2+z)

Ejercicios:
*a^2+ab=a(a+b)
*a^3+a+a=a(a^2+2)
*〖24a〗^2 xy^2-〖36x〗^2 y^4=12xy^2 (2a^2-3xy^2 )
*a^20-a^16+a^12-a^8+a^4-a^2=a^2 (a^18-a^14+a^10-a^6+a^2-1)

2.- FACTOR COMÚN POR AGRUPACION
ax+bx+ay+by=x(a+b)+y(a+b)=(x+y)(a+b)

Ejercicios:
*〖3m〗^2+6mn+4m-8n=3m(m-2n)+4(m-2n)=(3m+4)(m-2n)*〖2x〗^2-3xy-4x+6y=x(2x-3y)-2(2x-3y)=(x-2)(2x-3y)
*3ax-3x+4y-4ay=3x(a-1)-4y(-1+a)=(3x-4y)(a-1)
*ax-ay+az+x-y+z=a(x-y+z)+(x-y+z)=(a+1)(x-y+z)

3.- FACTOR COMÚN POR AGRUPACION
a^2+ab+a^2 =〖(a+b)〗^2

Ejercicios:
*〖4x〗^2+25y^2-20xy =〖4x〗^2-20xy+25y^2=〖(2x-5y)〗^2
*〖 y〗^2+bx+b^2/4=〖(x+b/2)〗^2
*〖 a〗^2+2a(a-b)+〖(a-b)〗^2=[a+(a-b)]^2=〖(2a-b)〗^2
*1/4-b/3+b^2/9=b^2/9-b/3+1/4=〖(b/3+1/2)〗^2

4.- DIFERENCIA DE CUADRADOS PERFECTOS
(a-b)^2=(a-b)(a+b)Ejercicios:
*〖 a〗^10-〖49 b〗^12=(a^2-7b^6 )(a^2+7b^6)
*〖 100m〗^2 〖 n〗^4-〖169 y〗^6=(10mn^2-7y^3 )(10mn^2+13y^3 )
*x^2/100-(y^2 z^4)/81=(x/10-(yz^2)/9)(x/10+(yz^2)/9)
*〖 100m〗^2 〖 n〗^4-〖1/16 x〗^8=(10mn^2-1/4 x^4 )(10mn^2+1/4 x^4 )

5.- TRINOMIO CUADRADO PERFECTOR POR ADICION Y SUSTRACCION:
(x^4+x^2 y^2+y^4+x^2 y^2)-x^2 y^2
(x^4+〖2x〗^2 y^2+y^4)-x^2 y^2
〖(x^2+y^2)〗^2-x^2 y^2〖(x^2+y^2)〗^2-〖(xy)〗^2
[(x^2+y^2 )-(xy)] [(x^2+y^2 )+(xy)]

Ejercicios:
*4a^4+8a^2 b^2+9b^4=(4a^4+8a^2 b^2+9b^4+16a^2 b^2)-〖16a〗^2 b^2
(2a^2+3b^2 )-〖(4ab)〗^2
(2a^2+3b^2-4ab)(2a^2+3b^2+4ab)
*a^4-16a^2 b^2+36b^4=〖(a〗^4-16a^2 b^2+36b^4++4a^2 b^2)-〖4a〗^2 b^2
〖〖(a〗^4+6b^4)〗^2-〖4a〗^2 b^2
〖〖(a〗^4+6b^4)〗^2-〖(2ab)〗^2
(a^4+6b^4-2ab)(a^4+6b^4+2ab)
*49m^4-151m^2 n^2+81n^8=(49m^4-151m^2 n^2+81n^8+25m^2 n^2)-25m^2n^2
(7m^2-9n^4 )-〖(5mn)〗^2
(7m^2-9n^4-5mn)(7m^2-9n^4+5mn)

6.- TRINOMIO DE LA FORMA x^2+bx+c:
x^2+5x+6=0
(x+3)(x+2)

Ejercicios:
*x^2-7x+6=(x+1)(x+6)
*a^2-13a+40=(a-8)(a-5)
*m^2-11m-12=(m-12)(m+1)
*n^2+28m-29=(n+29)(n-1)
*a^2-2a-15=(a-5)(a+3)

7.- TRINOMIO DE LA FORMA ax^2+bx+c:
〖15x〗^4-11x-12=0
(15x^2+20)(15x^2+9)/15=5(3x^2+4)3(5x^2+3)/15=(3x^2+4)(5x^2+3)

8.- CUBO DE UN BINOMIO(a+b)^3=(a^3+3a^2 b+3ab^2+b^3)
(a-b)^3=(a^3-3a^2 b+3ab^2-b^3)

Ejercicios:
(1+4a)^3=(1+12a+48a^2+64a^3)
(a^3-6b^5 )^3=(a^9-18a^6 b^5+108a^3 b^10-216b^15)

9.- CUBO PERFECTO
a^3+b^3=(a+b)(a^2-ab+b^2)
a^3-b^3=(a-b)(a^2+ab+b^2)

Ejercicios:
*a^3-8=(a-2)(a^2+2a+4)
*1-8x^3=(1-2x)(1+2x+4x^2)
*[(a+b)^3+1]=[(a+b)+1][(a+b)^2-(a+b)+1]

10.- SUMA Y DIFERENCIA DE POTENCIAS IGUALESa^5+b^5=(a+b)(a^4-a^3 b+a^2 b^2-ab^3+b^4)
a^7+b^7=(a+b)(a^6-a^5 b+a^4 b^2-a^3 b^3+a^2 b^4-ab^5+b^6)

Ejercicios:
* (x^7-1)/(x-1)=((x-1)(x^6+x^5+x^4+x^3+x^2+x+1))/(x-1)=(x^6+x^5+x^4+x^3+x^2+x+1)

* (m^5+n^5)/(m^(2+) n^2 )=((m+n)(m^4-m^3 n+m^2 n^2-mn^3+n^4))/(x-1)=(m^4-m^3 n+m^2 n^2-mn^3+n^4)

EJERCICIOS GENERALES:

*(2xy-2x+3-3)/(18x^3+15x^2-63x)=(2x(y-1)-3(y-1))/(3x(6x^2+5x-21))=((2x-3y)(y-1))/(3x(6x+14)(7x-9))=((2x-3y)(y-1))/(3x(3x+7)(2x-3))=(y-1)/(3x(3x+7))
* (3x^2-12x-x^2 y+4y)/(18x^3+15x^2-63x)=(3x(x^2-4)-y(x^2-4))/(3x-(6x^2+5x-21) )=(3x-y)(x^2-4)/( 3x(3x+7)(2x-3) )=((3x-y)(x-2)(x+2))/(3x(3x+7)(2x-3))
* ((x^2-y^2 )[x^4 y-(x^5 (x^2 y^6 )^(1⁄2) )^(1⁄3) ] )/((x-y)(x^3+x^2 y)-(x-y)/(x+y))^(-1) )= (x^2-y^2 )[x^4 y×-(x^5 (xy^3 ) )^(1⁄3) ]/((x-y)(x^3+x^2y)-(x-y)/(1/((x+y))))=((x^2-y^2 )[x^4 y-(x^6 y^3 )^(1⁄3) ] )/((x-y) x^2 (x+y)-(x-y)(x+y))=((x^2-y^2 )[x^4 y-x^2 y] )/((x-y)(x+y)(x^2-1))=((x+y)(x-y)(x^2 y)(x^2-1))/((x+y)(x-y)(x^2-1))=x^2 y
* (((1+x)/(1-x)-(1-x)/(1+x)) x^2)/((1+x)/(1-x)-1)(1-1/(1+x)) =((((1+x)(1+x)-(1-x)(1-x))/((1-x)(1+x))) x^2)/(((1+x)-(1-x))/(1-x))(((1+x)-1)/(1+x)) =((((1+2x+x^2)-(1-2x+x^2 ))/((1-x)(1+x))) x^2)/((1+x-1+x)/(1-x))(x/(1+x))=(((1+2x+x^2-1+2x-x^2)/((1-x)(1+x))) x^2)/(2x/(1-x))(x/(1+x)) =((4x/((1-x)(1+x))) x^2)/(((2x^2)/((1+x)(1-x))) )=((4x^3)/((1-x)(1+x)))/((2x^2)/((1+x)(1-x)))=(4x^3 (1+x)(1-x))/(2x^(2(1-x)(1+x)) )=(4x^3 (1+x)(1-x))/(2x^2 (1+x)(1-x))=2x

* ((x^2+x-6)/(x^2+4x+3))/((x^2-3x+2)/(x^2-1))=(((x+3)(x-2))/((x+3)(x+1)))/(((x-2)(x-1))/((x+1)(x-1)))=(((x-2))/((x+1)))/(((x-2))/((x+1) ))=((x-2)(x+1))/(x-2)(x+1)...