Question
Simplify the following expressions:
$(x^2 - x + 1)^2 - (x^2 + x + 1)^2$
$(x^2 - x + 1)^2 - (x^2 + x + 1)^2$
✓
Answer
Expanding,
we get $\left[x^2-x+1\right]^2-\left[x^2+x+1\right]^2$
$\left.=\left(x^2\right)^2+(-x)^2+1^2+2\left(x^2\right)(-x)+2(-x)(1)+2 x^2\right)-\left[\left(x^2\right)^2+x^2+1+2 x^2 x+2 x(1)+2 x^2(1)\right]$
${\left[\therefore(x+y+z)^2=x^2+y^2+z^2+2 x y+2 y z+2 x z\right]}$
$=x^4+y^2+1-2 x^3-2 x+2 x^2-x^2-x^4-1-2 x^3-2 x-2 x^2$
$=-4 x^3-4 x=-4 x\left(x^2+1\right)$
Hence simplified equation
$=\left[x^2-x+1\right]^2-\left[x^2+x+1\right]^2$
$=-4 x\left(x^2+1\right)$
we get $\left[x^2-x+1\right]^2-\left[x^2+x+1\right]^2$
$\left.=\left(x^2\right)^2+(-x)^2+1^2+2\left(x^2\right)(-x)+2(-x)(1)+2 x^2\right)-\left[\left(x^2\right)^2+x^2+1+2 x^2 x+2 x(1)+2 x^2(1)\right]$
${\left[\therefore(x+y+z)^2=x^2+y^2+z^2+2 x y+2 y z+2 x z\right]}$
$=x^4+y^2+1-2 x^3-2 x+2 x^2-x^2-x^4-1-2 x^3-2 x-2 x^2$
$=-4 x^3-4 x=-4 x\left(x^2+1\right)$
Hence simplified equation
$=\left[x^2-x+1\right]^2-\left[x^2+x+1\right]^2$
$=-4 x\left(x^2+1\right)$
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