Verified Answer :

  • Evaluate the numerator at \(t=3\) using the Remainder Theorem to find the remainder \(R=124\).
  • Divide the polynomial \(t^4+t^3+2t^2-t+1\) by \(t-3\) to find the quotient polynomial \(Q(t) = t^3 + 4t^2 + 14t + 41\).
  • Assemble the final expression as \(Q(t) + \frac{R}{t-3}\).
  • Final expression: \(\boxed{t^3 + 4t^2 + 14t + 41 + \frac{124}{t-3}}\).

Explanation

  1. Analyzing the Problem

We are given the rational expression \(\frac{t^4+t^3+2t^2-t+1}{t-3}\). To perform this division, we express the polynomial \(P(t) = t^4+t^3+2t^2-t+1\) as a quotient polynomial \(Q(t)\) plus a remainder term \(\frac{R}{t-3}\), where \(R\) is a constant. We can find this by using the Remainder Theorem and polynomial division. According to the Remainder Theorem, the remainder when \(P(t)\) is divided by \(t-3\) is equal to \(P(3)\).

  1. Calculating the Remainder

We calculate the remainder \(R\) by evaluating the polynomial \(P(t)\) at \(t=3\):
\[R = P(3) = (3)^4 + (3)^3 + 2(3)^2 - (3) + 1\]
\[R = 81 + 27 + 2(9) - 3 + 1\]
\[R = 81 + 27 + 18 - 3 + 1 = 124\]
Thus, the remainder is \(124\).

  1. Performing Polynomial Division

We perform polynomial long division to find the quotient \(Q(t)\). Dividing \(t^4+t^3+2t^2-t+1\) by \(t-3\):

  1. The leading term of the quotient is \(\frac{t^4}{t} = t^3\).
  2. Subtracting \(t^3(t-3) = t^4 - 3t^3\) from the original polynomial gives \((t^4+t^3+2t^2-t+1) - (t^4 - 3t^3) = 4t^3+2t^2-t+1\).
  3. The next term is \(\frac{4t^3}{t} = 4t^2\).
  4. Subtracting \(4t^2(t-3) = 4t^3 - 12t^2\) gives \((4t^3+2t^2-t+1) - (4t^3 - 12t^2) = 14t^2-t+1\).
  5. The next term is \(\frac{14t^2}{t} = 14t\).
  6. Subtracting \(14t(t-3) = 14t^2 - 42t\) gives \((14t^2-t+1) - (14t^2 - 42t) = 41t+1\).
  7. The last term is \(\frac{41t}{t} = 41\).
  8. Subtracting \(41(t-3) = 41t - 123\) gives \((41t+1) - (41t - 123) = 124\), which matches our remainder.

The quotient is \(Q(t) = t^3 + 4t^2 + 14t + 41\).

  1. Conclusion

Combining the quotient and the remainder, the result of the division is:
\[\frac{t^4+t^3+2t^2-t+1}{t-3} = t^3 + 4t^2 + 14t + 41 + \frac{124}{t-3}\]
This expresses the original expression as a quotient polynomial plus the remainder fraction. The final result is \(\boxed{t^3 + 4t^2 + 14t + 41 + \frac{124}{t-3}}\).

Examples

Polynomial division is a fundamental tool in engineering and computer science, particularly in control systems and signal processing. For instance, when analyzing the stability of a system, engineers often need to divide transfer functions (which are ratios of polynomials) to simplify complex systems into a series of easier-to-manage parts. Specifically, separating a polynomial fraction into a simpler polynomial plus a remainder helps in identifying the system's long-term behavior and transient responses, similar to how we decomposed this expression into a polynomial and a fractional remainder.

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