Using notable products develop the following algebraic expressions:
- $$(x+y+1)^2$$
- $$(x+1)\cdot(x-1)\cdot(x-1)\cdot(x+1)$$
- $$(x^2-2)^3$$
See development and solution
Development:
1) $$(x+y+1)^2=(x+(y+1))^2=x^2+2\cdot x\cdot(y+1)+(y+1)^2=$$
$$=x^2+2xy+2x+(y^2+2y+1)=x^2+y^2+2xy+2x+2y+1$$
2) $$(x+1)\cdot(x-1)\cdot(x-1)\cdot(x+1)=$$
$$=((x+1)\cdot(x-1))\cdot((x-1)\cdot(x+1))=$$
$$=(x^2-1)\cdot(x^2-1)=(x^2-1)^2=x^4-2x^2+1$$
3) $$(x^2-2)^3=(x^2)^3-3\cdot(x^2)^2\cdot2+3\cdot x^2\cdot 2^2-2^3=x^6-6x^4+12x^2-8$$
Solution:
- $$x^2+y^2+2xy+2x+2y+1$$
- $$x^4-2x^2+1$$
- $$x^6-6x^4+12x^2-8$$