Find the remainder when $ t^3 + 2t^2 - 5t + 6 $ is divided by $ t - 1 $. - Silent Sales Machine

Understanding the Context

Finding the Remainder When $ t^3 + 2t^2 - 5t + 6 $ Is Divided by $ t - 1 $

When dividing a polynomial by a linear divisor of the form $ t - a $, the Remainder Theorem provides an efficient way to find the remainder without performing full polynomial long division. This theorem states:

> Remainder Theorem: The remainder of the division of a polynomial $ f(t) $ by $ t - a $ is equal to $ f(a) $.

In this article, we’ll apply the Remainder Theorem to find the remainder when dividing:
$$
f(t) = t^3 + 2t^2 - 5t + 6
$$
by $ t - 1 $.

Key Insights

Step 1: Identify $ a $ in $ t - a $

Here, the divisor is $ t - 1 $, so $ a = 1 $.

Step 2: Evaluate $ f(1) $

Substitute $ t = 1 $ into the polynomial:
$$
f(1) = (1)^3 + 2(1)^2 - 5(1) + 6
$$
$$
f(1) = 1 + 2 - 5 + 6 = 4
$$

Step 3: Interpret the result

By the Remainder Theorem, the remainder when $ t^3 + 2t^2 - 5t + 6 $ is divided by $ t - 1 $ is $ oxed{4} $.

This method saves time and avoids lengthy division—especially useful in algebra, calculus, and algebraic modeling. Whether you're solving equations, analyzing functions, or tackling polynomial identities, the Remainder Theorem simplifies key calculations.

Real-World Applications

Understanding polynomial remainders supports fields like engineering, computer science, and economics, where polynomial approximations and function evaluations are essential. For example, engineers use remainder theorems when modeling system responses, while data scientists apply polynomial remainder concepts in regression analysis.

Final Thoughts

Conclusion:
To find the remainder of $ t^3 + 2t^2 - 5t + 6 $ divided by $ t - 1 $, simply compute $ f(1) $ using the Remainder Theorem. The result is 4—fast, accurate, and efficient.

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