domingo, 10 de octubre de 2010

Ejercicios de Matemáticas






PROPIEDADES DE LOS NUMEROS
(Escalera)

 1.)\:\qquad x(y+z+w)=xy+xz+xw
                                                

2.)\:\qquad (a+b)(c+d)=a(c+d)+b(c+d)
                    \:\qquad = ac+ad+bc+bd

3.)\:\qquad (7x-3y)^2=(7x-3y)(7x-3y)

\qquad\qquad=7x(7x-3y)-3y(7x-3y)

\qquad\qquad=49x^2-21xy-21yx+9y^2

\qquad\qquad=49x^2-42xy+9y^2            
                

 4.)\:\qquad 5(3x-y)+5(x+5y)-4(2x+y)= 

\qquad\qquad=15x-5y+5x+25y-8x-4y

\qquad\qquad=12x+16y

5.)\:\qquad (7a-2b)(a^2-5ab+2b^2)=

\qquad\qquad=7a(a^2-5ab+2b^2)-2b(a^2-5ab+2b^2)

\qquad\qquad=7a^3-35a^2b+14ab^2-2ba^2+10ab^2-4b^3

\qquad\qquad=7a^3-37a^2b+24ab^2-ab^3

6.)\:\qquad (m^2-m(4m+2n)+n(5-n(m+1)-(7m^2+mn-3n^2)))

\qquad\qquad=(m^2-m(4m2n)+n(5-nm-n-7m^2-mn+3^2))

\qquad\qquad=(m^3-4m^2-2nm+5n-n^2m-n^2-7m^2n-mn^2+3n^2)

\qquad\qquad=m^3-4m^2-2mn+5n-2n^2m+2n^2-7m^2n


SINTAXIS Y SEMANTICA
(Escalera)
1.)\:\qquad ab+c










2.)\:\qquad abc+d













3.)\:\qquad 6x^2-8x+1
 












4.)\:\qquad 7+sen^3{x^2}


















POLINOMIOS
(Escalera)
1.)\:\qquad x^2-3x^2+7x+4

2.)\:\qquad 60x^2-85x-24

3.)\:\qquad xy-3z+2

4.)\:\qquad \frac{12m^2}{n}+\frac{17m}{\sqrt{n}}-14\sqrt{n}

5.)\:\qquad 45x^7+35x^5-8x^3

6.)\:\qquad \frac{3u^3v^4-2u^5v^2+(u^2v^2)^2}{u^3v^2}

                             
ECUACIONES

(Escalera) 
1.)\:\qquad 6z-7=2z+5\qquad\qquad z=3

\qquad\qquad 6(3)-7=2(3)+5

\qquad\qquad 18-7=6+5

\qquad\qquad 11=11
2.)\:\qquad (3x-4)-9x=6x+8     \qquad\qquad x=-1

\qquad\qquad (3(-1)-4)-9(-1)=6(-1)+8

\qquad\qquad (-3-4)+9 = -6 +8

\qquad\qquad -7+9=2

\qquad\qquad 2=2

3.)\:\qquad (x+3)-(3x-1)=0      \qquad\qquad x=2

\qquad\qquad ((2)+3)-(3(2)-1)=0

\qquad\qquad (5)-(6-1)=0

\qquad\qquad 5-5=0

\qquad\qquad 0=0

4.)\:\qquad 5x+3=7x-2       \qquad\qquad x=\frac{5}{2}
        \qquad\qquad 5(\frac{5}{2})+3=7(\frac{5}{2})-2
        \qquad\qquad (\frac{25}{2})+3=(\frac{35}{2})-2
                         \frac{31}{2}=\frac{31}{2}


 5)\:\qquad 12w-7w=2w+1 \qquad w=\frac{1}{3}
                        12(\frac{1}{3}) - 7 (\frac{1}{3}) = 2 (\frac{1}{3}) + 1
                       (\frac{12}{3}) - (\frac{7}{3}) =  (\frac{2}{3}) + 1
                        \frac{5}{3} =  \frac{5}{3}


OPERACIONES ALGEBRAICAS SIMPLIFICAR
(Escalera)


 1)\:\qquad 3a+(2+5a)

\qquad\qquad 3a + 2 + 5a

\qquad\qquad 8a +2
 2)\:\qquad 2x-(2-x)

\qquad\qquad 2x-2+x

\qquad\qquad 3x-2
 3)\:\qquad 5x+ [6-(2x-1)]

\qquad\qquad  5x + [6-2x+1]

\qquad\qquad 5x + 6 - 2x + 1

\qquad\qquad 3x + 7
 4)\:\qquad 3y-[x-2(3x-y)] - [2y-(x+3y)]

\qquad\qquad  3y-[ x -6x + 2y] - [2y-x-3y]

\qquad\qquad  3y-x+6x-2y-2y+x+3y

\qquad\qquad 2y +6x
 5)\:\qquad 15-5[4-2(x+1)]-[3x-5(x+4)]

\qquad\qquad  15-5[4-2x-2]-[3x-5x-20]

\qquad\qquad  15 +20+10x+10 -3x+5x+20

\qquad\qquad 12x +65

ALGEBRA - SIMPLIFICAR
(Escalera)
 1)\:\qquad (-2^2ab^4)^3(a^2b)^5

\qquad\qquad  (64a^3b^1^2) (a^1^0b^5)

\qquad\qquad  64 a^1^3b^1^7

\qquad\qquad
 2)\:\qquad (x^2)(x^3)-(-x^2)(x)

\qquad\qquad  x^5+x^3

\qquad\qquad  

\qquad\qquad
 3)\:\qquad 3x^2(x^3-2x^2+1)

\qquad\qquad  3x^5-6x^4+3x^2

\qquad\qquad  

\qquad\qquad
 4)\:\qquad (2ab^3)^2(-3^2a^2c)^3(-a^4bc^2)^5

\qquad\qquad  (4a^2^6)(729 a^6c^3)(-a^2^0b^5c^1^0)

\qquad\qquad  -2916 a^5^2b^5c^1^3

\qquad\qquad

 5)\:\qquad -2a^2b(a^3+5a^2b^2-3b^4)

\qquad\qquad  -2a^5b -10a^4b^3+6a^2b^4

\qquad\qquad  

\qquad\qquad
 MULTIPLICACION DE POLINOMIOS
(Escaleras)
 1)\:\qquad (x+7)(x-3)

\qquad\qquad x(x-3)+7(x-3)

\qquad\qquad x^2 - 3x +7x-21

\qquad\qquad x^2+4x-21
 2)\:\qquad (2x-5)(x+7)

\qquad\qquad 2x(x+7)-5(x+7)

\qquad\qquad 2x^2 +14x-5x-35

\qquad\qquad 2x^2 +9x-35
 3)\:\qquad (x+2)(x-4) -x(x-2)

\qquad\qquad x(x-4)+2(x-4)-x(x-2)

\qquad\qquad x^2-4x+2x-8-x^2+2x

\qquad\qquad  -8

 4)\:\qquad (2x-1)(4x^2+2x+1)

\qquad\qquad 2x(4x^2+2x+1)-1(4x^2+2x+1)

\qquad\qquad 8x^3+4x^2+2x-4x^2-2x-1

\qquad\qquad  8x^3-1
 5)\:\qquad (x-2y)(x^2+2xy+4y^2)

\qquad\qquad x(x^2+2xy+4y^2)-2y(x^2+2xy+4y^2)

\qquad\qquad x^3 +2x^2y +4y^2x -2yx^2-4xy^2-8y^3

\qquad\qquad x^3 -8y^3
EFECTUAR OPERACIONES CON FRACCIONES
(Escaleras)
 1)\:\qquad (\frac{6x^2y^3z}{8xy^3z^2})^3

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
                      \qquad\qquad (\frac{216x^6y^9z^3}{512x^3y^9z^6})

\qquad\qquad 

\qquad\qquad
                      \qquad\qquad (\frac{216x^3}{512z^3})

\qquad\qquad 

\qquad\qquad

 2)\:\qquad \frac{-44a^3b^2}{66a^5b^8}

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
                       \:\qquad \frac{-44}{66a^2b^6}

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
 3)\:\qquad \frac{x^6y^4}{x^3y^2}

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
                       \:\qquad x^3y^2

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
FACTORIZACION POR FACTOR COMUN
(Escaleras)
 1)\:\qquad 7xy+2xz

\qquad\qquad x(7y+2z)

\qquad\qquad 

\qquad\qquad
 2)\:\qquad 4xy-6xz+8xw

\qquad\qquad 2x(2y-3z+4w)

\qquad\qquad 

\qquad\qquad
 3)\:\qquad 5a^5b^4-10a^3b^6+20a^2b^8

\qquad\qquad 5a^2b^4(a^3-2ab^2+4b^4)

\qquad\qquad 

\qquad\qquad
 4)\:\qquad x^3-9x^2+2x-18

\qquad\qquad x^2(x-9)+2(x-9)

\qquad\qquad (x-9)(x^2+2)

\qquad\qquad
 5)\:\qquad x^3+4x^2y+xy^2+4y^2

\qquad\qquad x^3+xy^2+4x^2y+4y^2

\qquad\qquad xy(x^2+y)+4y(x^2+y)

\qquad\qquad (x^2+y)(xy+4y)
 6)\:\qquad 7x^2-14x-6x+12

\qquad\qquad 7x(x-2)-6(x-2)

\qquad\qquad (x-2)(7x-6)

\qquad\qquad
FACTORIZACION DE TRINOMIOS
(Escaleras)
 1)\:\qquad x^2-x-6

\qquad\qquad (x+2)(x-3)

\qquad\qquad 

\qquad\qquad




 2)\:\qquad x^2+2x-8

\qquad\qquad (x-2)(x+4)

\qquad\qquad 

\qquad\qquad
 3)\:\qquad  x^2-4x-21

\qquad\qquad (x+3)(x-7)

\qquad\qquad 

\qquad\qquad
 4)\:\qquad  5x^2-14x-3

\qquad\qquad (x-3)(5x+1)

\qquad\qquad 

\qquad\qquad
 5)\:\qquad  5x^2-17x+6

\qquad\qquad (x-3)(5x-2)

\qquad\qquad 

\qquad\qquad
 6)\:\qquad  7x^2-20x+12

\qquad\qquad (x-2)(7x-6)

\qquad\qquad 

\qquad\qquad

FACTORIZACION CON FRACCIONES
(Escaleras)


 1)\:\qquad  \frac{9x^2-6xy-12y^2}{3xy}

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
          \:\qquad  \frac{3x}{y} - \frac{2}{1}-\frac{6y}{x}

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
 2)\:\qquad  \frac{2a^4b-4a^3b^2+2a^2b^3}{2a^2b}

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
         \:\qquad  \frac{a^2-2ab+b^2}{1} 

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
         \:\qquad  \frac{(a-b)(a+b)}{1} 

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
 3)\:\qquad  \frac{a^3b^3-2a^2b^4-15ab^5}{ab^3}

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
         \:\qquad  \frac{a^2-2ab-15b^2}{1} 

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
         \:\qquad  \frac{(a-5b)(a+3b)}{1} 

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
        
 4)\:\qquad  \frac{6x^2-7x-5}{3x^2-2x-5}

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
         \:\qquad  \frac{(2x+1)(3x-5)}{(3x-5)(x+1)} 

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
         \:\qquad  \frac{2x+1}{x+1} 

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
 5)\:\qquad  \frac{12y^2+3y}{20y^2+9y+1}

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
         \:\qquad  \frac{3y(4y+1)}{(5y+1)(4y+1)} 

\qquad\qquad 

\qquad\qquad 

\qquad\qquad
         \:\qquad  \frac{3y}{5y+1} 

\qquad\qquad 

\qquad\qquad 

\qquad\qquad

FORMULA GENERAL

 ax^2+bx+c=0
 (\frac{1}{a})(ax^2+bx+c)=(0)(\frac{1}{a})
 x^2+\frac{b}{a}x=-\frac{c}{a}
 x^2+\frac{b}{a}x+(\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2
 x^2+\frac{b}{a}x+\frac{b^2}{4a^2} =-\frac{c}{a}+\frac{b^2}{4a^2}
 (x+\frac{b}{2a})^2 =\frac{b^2-4ac}{4a^2}
 \sqrt{(x+\frac{b}{2a})^2} = \frac{+}{}\sqrt{\frac{b^2-4ac}{4a^2}}
 x+\frac{b}{2a} = \frac{+}{}\sqrt{\frac{b^2-4ac}{4a^2}}
 x =-\frac{b}{2a} \frac{+}{}\sqrt{\frac{b^2-4ac}{4a^2}}
 x =-\frac{b}{2a}+\frac{1}{2a} \frac{+}{}\sqrt{b^2-4ac}
 x =\frac{-b\frac{+}{}\sqrt{b^2-4ac}}{2a}