Question

Factorizar x²+5×+6 , 1+2a³+a⁶

Answer 1

Respuestas:

1) $\mathrm{Factorizar}\:x^2+5x+6:\quad \left(x+2\right)\left(x+3\right)$

2) $\mathrm{Factorizar}\:1+2a^3+a^6:\quad \left(a+1\right)^2\left(a^2-a+1\right)^2$

Pasos:

1) $x^2+5x+6$

$\mathrm{Factorizar\:la\:expresion}$

$=\left(x^2+2x\right)+\left(3x+6\right)$

$\mathrm{Factorizar\:}x\mathrm{\:de\:}x^2+2x\mathrm{:\quad }x\left(x+2\right)$

$\mathrm{Factorizar\:}3\mathrm{\:de\:}3x+6\mathrm{:\quad }3\left(x+2\right)$

$=x\left(x+2\right)+3\left(x+2\right)$

$\mathrm{Factorizar\:el\:termino\:comun\:}x+2$

$=\left(x+2\right)\left(x+3\right)$

2) $1+2a^3+a^6$

$\mathrm{Utilizar\:el\:teorema\:de\:la\:raiz\:racional}$

$a=1,\:\quad a_n=1$

$\mathrm{Los\:divisores\:de\:}a:\quad 1,\:\quad \mathrm{Los\:divisores\:de\:}a_n:\quad 1$

$\mathrm{Por\:lo\:tanto,\:verificar\:los\:siguientes\:numeros\:racionales:\quad }\pm \frac{1}{1}$

$\frac{1}{1}\mathrm{\:es\:la\:raiz\:de\:la\:expresion,\:por\:lo\:tanto,\:factorizar\:}a+1$

$=\left(a+1\right)\frac{a^6+2a^3+1}{a+1}$

$\frac{a^6+2a^3+1}{a+1}=a^5-a^4+a^3+a^2-a+1$

$=a^5-a^4+a^3+a^2-a+1$

$\mathrm{Factorizar}\:a^5-a^4+a^3+a^2-a+1:\quad \left(a^2-a+1\right)\left(a+1\right)\left(a^2-a+1\right)$

$=\left(a+1\right)\left(a^2-a+1\right)\left(a+1\right)\left(a^2-a+1\right)$

$\mathrm{Simplificar}:$

$=\left(a+1\right)^2\left(a^2-a+1\right)^2$