Problems from Notable products

Using notable products develop the following algebraic expressions:

  1. $$(x+y+1)^2$$
  2. $$(x+1)\cdot(x-1)\cdot(x-1)\cdot(x+1)$$
  3. $$(x^2-2)^3$$
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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:

  1. $$x^2+y^2+2xy+2x+2y+1$$
  2. $$x^4-2x^2+1$$
  3. $$x^6-6x^4+12x^2-8$$
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