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Question
Simplify the following expressions:
`(x^2 - x + 1)^2 - (x^2 + x + 1)^2`
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Solution
We have,
`[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 (1)] - [(x^2)^2 + (x)^2 + (1)^2 + 2x^2 (x) + 2(x)(1) + 2(x^2)(1)]`
`= x^4 + x^2 + 1 - 2x^3 - 2x + 2x^2 - x^2 - x^4 - 1 - 2x^3 - 2x - 2x^2`
`[∵ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca]`
`= -4x^3 - 4x`
`= -4x [x^2 + 1]`
`∴ [x^2 - x + 1]^2 - [x^2 + x + 1]^2 = -4x[x^2 + 1]`
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