Simplify: \(\frac{x^4 + x^2 + 1}{x^2 + x + 1}\)
x2-x+1
Write the numerator as a difference of squares: \(x^4 + x^2 + 1 = (x^4 + 2x^2 + 1) - x^2 = (x^2 + 1)^2 - x^2\).
Using \(a^2 - b^2 = (a + b)(a - b)\), \(x^4 + x^2 + 1 = (x^2 + x + 1)(x^2 - x + 1)\).
Therefore \(\frac{x^4 + x^2 + 1}{x^2 + x + 1} = \frac{(x^2 + x + 1)(x^2 - x + 1)}{x^2 + x + 1} = x^2 - x + 1\).
Check: at x = 2, the expression = (16 + 4 + 1)/(4 + 2 + 1) = 21/7 = 3, and x2 – x + 1 = 4 – 2 + 1 = 3. The values agree.
Hence, x2-x+1.
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