Razonamiento Matematico: octubre 2010

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

$(\frac{1}{a})(ax^2+bx+c)=(0)(\frac{1}{a})$

$x^2+\frac{b}{a}x=-\frac{c}{a}$

$x^2+\frac{b}{a}x+(\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2$

$x^2+\frac{b}{a}x+\frac{b^2}{4a^2} =-\frac{c}{a}+\frac{b^2}{4a^2}$

$(x+\frac{b}{2a})^2 =\frac{b^2-4ac}{4a^2}$

$\sqrt{(x+\frac{b}{2a})^2} = \frac{+}{}\sqrt{\frac{b^2-4ac}{4a^2}}$

$x+\frac{b}{2a} = \frac{+}{}\sqrt{\frac{b^2-4ac}{4a^2}}$

$x =-\frac{b}{2a} \frac{+}{}\sqrt{\frac{b^2-4ac}{4a^2}}$

$x =-\frac{b}{2a}+\frac{1}{2a} \frac{+}{}\sqrt{b^2-4ac}$

$x =\frac{-b\frac{+}{}\sqrt{b^2-4ac}}{2a}$

Razonamiento Matematico

domingo, 10 de octubre de 2010

Ejercicios de Matemáticas

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