where \( R(t) \) is the remainder and has degree less than 2. Assume \( R(t) = at + b \).
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- ---**Question: A volcanologist is analyzing the seismic waves produced by a volcanic eruption. Let \( f(t) \) be a polynomial representing the amplitude of a seismic wave over time \( t \). Find the remainder when \( t^4 + 3t^2 + 2 \) is divided by \( t^2 + 1 \).
- To find the remainder when \( t^4 + 3t^2 + 2 \) is divided by \( t^2 + 1 \), we use polynomial long division or the remainder theorem. We express \( t^4 + 3t^2 + 2 \) in the form:
- t^4 + 3t^2 + 2 = (t^2 + 1)Q(t) + R(t)
- Divide the leading term \( t^4 \) by \( t^2 \) to get \( t^2 \).
- Multiply \( t^2 \) by \( t^2 + 1 \) to get \( t^4 + t^2 \).
- Subtract: \( (t^4 + 3t^2 + 2) - (t^4 + t^2) = 2t^2 + 2 \).
