the remainder obtained when t^7+3t^2+2t-6 divided by t-1

To find the remainder when \( t^7 + 3t^2 + 2t - 6 \) is divided by \( t - 1 \), we can use the **Remainder Theorem**. The Remainder Theorem states that if a polynomial \( f(t) \) is divided by \( t - a \), the remainder is \( f(a) \).

Given:
\[
f(t) = t^7 + 3t^2 + 2t - 6
\]
We want to find the remainder when this polynomial is divided by \( t - 1 \), so we substitute \( t = 1 \) into \( f(t) \):

\[
f(1) = 1^7 + 3(1^2) + 2(1) - 6
\]

Simplifying:

\[
f(1) = 1 + 3(1) + 2(1) - 6
\]
\[
f(1) = 1 + 3 + 2 - 6 = 0
\]

So, the remainder is \( 0 \).

### Conclusion:
The remainder when \( t^7 + 3t^2 + 2t - 6 \) is divided by \( t - 1 \) is **0**.

Would you like more details on this, or do you have any questions?

### Related Questions:
1. What is the remainder when \( t^7 + 3t^2 + 2t - 6 \) is divided by \( t + 1 \)?
2. Can you use synthetic division to confirm the result?
3. How would the remainder change if the divisor were \( t + 2 \) instead of \( t - 1 \)?
4. What is the degree of the remainder when dividing \( t^7 + 3t^2 + 2t - 6 \) by a quadratic polynomial?
5. How does the Remainder Theorem relate to the Factor Theorem?

### Tip:
When using the Remainder Theorem, remember that it only gives the remainder, not the quotient. For finding the quotient, you would need to perform polynomial division or synthetic division.

Learn how to find the remainder when dividing the polynomial t^7 + 3t^2 + 2t - 6 by t - 1 using the Remainder Theorem. Detailed step-by-step solution provided.

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