Question

son ejercicios de factorización quien me ayuda con los procedimientos por favor​

Answer 1

Respuesta:

$1)3ab^2\left(a+5-15b\right)$

$2)\left(x+3\right)\left(b+5\right)$

$3)\left(x-1\right)\left(a-1\right)$

$4)\left(x+3\right)\left(x+4\right)$

$5)\left(x+3\right)\left(x-5\right)$

$6)\left(4x+3y\right)\left(4x-3y\right)$

$7)\left(5a+2\right)\left(25a^2-10a+4\right)$

Explicación paso a paso:

$3a^2b^2+15ab^2-45ab^3\\\\\mathrm{Aplicar\:las\:leyes\:de\:los\:exponentes}:\quad \:a^{b+c}=a^ba^c\\\\ab^3=ab^2b,\:a^2b^2=aab^2\\\\=3aab^2+15ab^2-45ab^2b\\\\\mathrm{Reescribir\:}-45\mathrm{\:como\:}15\cdot \:3\\\\\mathrm{Reescribir\:}15\mathrm{\:como\:}5\cdot \:3\\\\=3aab^2+5\cdot \:3ab^2+5\cdot \:3\cdot \:3ab^2b\\\\\mathrm{Factorizar\:el\:termino\:comun\:}3ab^2\\\\=3ab^2\left(a+5-15b\right)$

$15+5x+3b+xb\\\\=\left(5x+15\right)+\left(xb+3b\right)\\\\\mathrm{Factorizar\:}b\mathrm{\:de\:}xb+3b\mathrm{:\quad }b\left(x+3\right)\\\\\mathrm{Factorizar\:}5\mathrm{\:de\:}5x+15\mathrm{:\quad }5\left(x+3\right)\\\\=b\left(x+3\right)+5\left(x+3\right)\\\\\mathrm{Factorizar\:el\:termino\:comun\:}x+3\\\\=\left(x+3\right)\left(b+5\right)$

$a\left(x-1\right)+1-x\\\\\mathrm{Expandir}\:a\left(x-1\right):\quad ax-a\\\\=ax-a+1-x\\\\\mathrm{Factorizar\: el\: polinomio}:\\\\=\left(x-1\right)\left(a-1\right)$

$x^2+7x+12\\\\=\left(x^2+3x\right)+\left(4x+12\right)\\\\\mathrm{Factorizar\:}x\mathrm{\:de\:}x^2+3x\mathrm{:\quad }x\left(x+3\right)\\\\\mathrm{Factorizar\:}4\mathrm{\:de\:}4x+12\mathrm{:\quad }4\left(x+3\right)\\\\=x\left(x+3\right)+4\left(x+3\right)\\\\\mathrm{Factorizar\:el\:termino\:comun\:}x+3\\\\=\left(x+3\right)\left(x+4\right)$

$x^2-2x-15\\\\=\left(x^2+3x\right)+\left(-5x-15\right)\\\\\mathrm{Factorizar\:}x\mathrm{\:de\:}x^2+3x\mathrm{:\quad }x\left(x+3\right)\\\\\mathrm{Factorizar\:}-5\mathrm{\:de\:}-5x-15\mathrm{:\quad }-5\left(x+3\right)\\\\=x\left(x+3\right)-5\left(x+3\right)\\\\\mathrm{Factorizar\:el\:termino\:comun\:}x+3\\\\=\left(x+3\right)\left(x-5\right)$

$16x^2-9y^2\\\\\mathrm{Reescribir\:}16x^2-9y^2\mathrm{\:como\:}\left(4x\right)^2-\left(3y\right)^2\\\\=\left(4x\right)^2-\left(3y\right)^2\\\\\mathrm{Aplicar\:la\:siguiente\:regla\:para\:binomios\:al\:cuadrado:\:}x^2-y^2=\left(x+y\right)\left(x-y\right)\\\\\left(4x\right)^2-\left(3y\right)^2=\left(4x+3y\right)\left(4x-3y\right)\\\\=\left(4x+3y\right)\left(4x-3y\right)$

$8+125a^3\\\mathrm{Reescribir\:}125a^3+8\mathrm{\:como\:}\left(5a\right)^3+2^3\\\\=\left(5a\right)^3+2^3\\\\\mathrm{Aplicar\:la\:siguiente\:regla\:de\:productos\:notables\:\left(Suma\:de\:cubos\right):\:}x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)\\\\\left(5a\right)^3+2^3=\left(5a+2\right)\left(5^2a^2-2\cdot \:5a+2^2\right)\\\\=\left(5a+2\right)\left(5^2a^2-2\cdot \:5a+2^2\right)\\\\=\left(5a+2\right)\left(25a^2-10a+4\right)$