To find the remainder of \( v^4 + 3v + 1 \) when divided by \( v^2 + 1 \), express the division as:
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- = \left( 8 - \frac{16}{3} \right) = \frac{24}{3} - \frac{16}{3} = \frac{8}{3}.
- \boxed{\frac{8}{3}}.
- Find the remainder when \( v^4 + 3v + 1 \) is divided by \( v^2 + 1 \).
- v^4 + 3v + 1 = (v^2 + 1)Q(v) + R(v),
- where \( R(v) \) is the remainder and has degree less than 2, so let \( R(v) = av + b \).
- Divide the leading term \( v^4 \) by \( v^2 \), giving \( v^2 \).
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