Question
Simplify the following expressions: $(x^2 - x + 1)^2 - (x^2 + x + 1)^2$
✓
Answer
Expanding,
we get $[x^2 - x + 1]^2 - [x^2 + x + 1]^2$
$= (x^2)^2 + (-x)^2 + 1^2 + 2(x^2)(-x) + 2(-x)(1) + 2x^2) - [(x^2)^2 + x^2 + 1 + 2x^2x + 2x(1) + 2x^2(1)]$
$\big[\therefore$$(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2xz$$\big]$
$= x^4+ y^2 + 1 - 2x^3 − 2x + 2x^2 - x^2 - x^4 - 1 - 2x^3 - 2x - 2x^2$
$= -4x^3 - 4x = -4x(x^2 + 1)$
Hence simplified equation
$= [x^2 - x + 1]^2 - [x^2 + x + 1]^2 = -4x(x^2 + 1)$
we get $[x^2 - x + 1]^2 - [x^2 + x + 1]^2$
$= (x^2)^2 + (-x)^2 + 1^2 + 2(x^2)(-x) + 2(-x)(1) + 2x^2) - [(x^2)^2 + x^2 + 1 + 2x^2x + 2x(1) + 2x^2(1)]$
$\big[\therefore$$(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2xz$$\big]$
$= x^4+ y^2 + 1 - 2x^3 − 2x + 2x^2 - x^2 - x^4 - 1 - 2x^3 - 2x - 2x^2$
$= -4x^3 - 4x = -4x(x^2 + 1)$
Hence simplified equation
$= [x^2 - x + 1]^2 - [x^2 + x + 1]^2 = -4x(x^2 + 1)$
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