To find \( a \) and \( b \), perform polynomial division or use substitution with roots of the divisor.
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- We are to divide \( u^4 + 3 \) by \( u^2 - 2u + 2 \) and find the remainder. Since the divisor is quadratic, the remainder must be linear or constant: let \( R(u) = au + b \).
- We write:
- u^4 + 3 = (u^2 - 2u + 2)Q(u) + au + b
- Letâs use the roots of \( u^2 - 2u + 2 = 0 \). The discriminant is \( 4 - 8 = -4 \), so roots are:
- u = rac{2 \pm \sqrt{-4}}{2} = rac{2 \pm 2i}{2} = 1 \pm i
- Let \( lpha = 1 + i \), \( eta = 1 - i \). Then at \( u = lpha \):
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