Ejercicios Resueltos de Factorización

a)  $x^2-2xy+y^2$→$(x-y)^2\bullet$

b) $x^2-100$→$x^2-10^2$→$(x+10)(x-10)\bullet$

c)  $x^2+7x+10$→$(x+5)(x+2)\bullet$

d)  $x^2-x+1/4$→$(x-1/2)(x-1/2)\bullet$

e)  $2x^2-x-3$→$\displaystyle\frac{2(2x^2-x-3)}{2}$→$\displaystyle\frac{(2x)^2-(2x)-6)}{2}$
→$\displaystyle\frac{(2x-3)(\color{red}{2}x+\color{red}{2})}{\color{red}{2}}$→$(2x-3)(x+1)\bullet$

f)  $16z^3y^2-24z^2y^2+20z^2y^3$
→$4z.\color{red}{4.z^2y^2}-6.\color{red}{4.z^2y^2}+5.\color{red}{4.z^2y^2}y$→$4z^2y^2(4z-6+5y)\bullet$

g)  $2x^2-3xy-4x+6y$→$2x^2-4x+6y-3xy$
→$\color{red}{2x}.x-2.\color{red}{2x}+2.\color{red}{3y}-x.\color{red}{3y}$→$2x(x-2)+3y(2-x)$
→$2x\color{red}{(x-2)}-3y\color{red}{(x-2)}$→$(x-2)(2x-3y)\bullet$

h)  $a^{2n}-b^{2n}$→$(a^n)^2-(b^n)^2$→$(a^n+b^n)(a^n-b^n)\bullet$

i)  $(2x+3)^3-(2x-3)^3$→$\color{red}{a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)}$
→$\left[2x+3-(2x-3)\right] \left[(2x+3)^2+(2x+3)(2x-3)+(2x-3)^2\right]$
→$(2x+3-2x+3)(4x^2+12x+9+4x^2-9+4x^2-12x+9)$
→$6(12x^2+9)$→$18(4x^2+3)\bullet$

j)  $6x^2-7x-5$→$\displaystyle\frac{6(6x^2-7x-5)}{6}$
→$\displaystyle\frac{(6x)^2-7(6x)-30)}{6}$
→$\displaystyle\frac{(6x-10)(6x+3)}{2.3}$→$(3x-5)(2x+1)\bullet$

k)  $10x^2-9x+2$→$\displaystyle\frac{10(10x^2-9x+2)}{10}$→$\displaystyle\frac{(10x)^2-9(10x)+20)}{10}$
→$\displaystyle\frac{(10x-5)(10x-4)}{2.5}$→$(2x-1)(5x-2)\bullet$