Solution: By the Remainder Theorem, the remainder when $ A(x)^3 + 2 $ is divided by $ x - 5 $ is $ A(5)^3 + 2 $. So we must compute $ A(5) $. - Appfinity Technologies

Understanding the Context

Understanding the Remainder Theorem in Polynomial Division

When dividing a polynomial $ f(x) $ by a linear divisor $ x - c $, the Remainder Theorem states that the remainder is simply $ f(c) $. This powerful and concise method eliminates the need for long division in many cases.

In this article, we explore how this theorem applies when $ f(x) = A(x)^3 + 2 $, and how computing $ A(5) $ allows us to find the remainder without expanding the full polynomial expression.

Key Insights

Step-by-Step: Applying the Remainder Theorem

Let’s analyze what happens when dividing $ A(x)^3 + 2 $ by $ x - 5 $:

1. By the Remainder Theorem, the remainder is:
   $$
   f(5) = A(5)^3 + 2
   $$
   
2. This means we can directly compute $ A(5) $ — a crucial step that avoids unnecessary expansion of $ A(x)^3 + 2 $.
   
3. The full computation of $ A(5) $ depends on the definition of $ A(x) $. For example, if $ A(x) $ is a simple polynomial like $ A(x) = x $ or $ A(x) = 2x + 1 $, substituting $ x = 5 $ gives a quick result.

But even for complex $ A(x) $, knowing $ A(5) $ turns an abstract division problem into a simple numerical evaluation.

Final Thoughts

Why Computing $ A(5) $ Matters

Computing $ A(5) $ serves two key purposes:

• Reduces complexity: Instead of expanding $ A(x)^3 + 2 $ into a lengthy polynomial, we reduce the problem to a single evaluation.
  
• Enables direct remainder calculation: Only after computing $ A(5) $ do we plug it into $ A(5)^3 + 2 $ to get the exact remainder of the division.

Example: A Clear Illustration

Suppose $ A(x) = 2x + 1 $. Then: