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) $. - IQnection
Title: How to Find the Remainder of $ A(x)^3 + 2 $ Divided by $ x - 5 $ Using the Remainder Theorem
Title: How to Find the Remainder of $ A(x)^3 + 2 $ Divided by $ x - 5 $ Using the Remainder Theorem
Meta Description:
Learn how to compute the remainder when $ A(x)^3 + 2 $ is divided by $ x - 5 $ using the Remainder Theorem. Discover the role of $ A(5) $ in simplifying the expression and efficiently finding the result.
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.
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Key Insights
Step-by-Step: Applying the Remainder Theorem
Let’s analyze what happens when dividing $ A(x)^3 + 2 $ by $ x - 5 $:
- By the Remainder Theorem, the remainder is:
$$
f(5) = A(5)^3 + 2
$$ - This means we can directly compute $ A(5) $ — a crucial step that avoids unnecessary expansion of $ A(x)^3 + 2 $.
- 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.
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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:
- Evaluate $ A(5) = 2(5) + 1 = 11 $
- Plug into $ A(x)^3 + 2 $:
$$
A(5)^3 + 2 = 11^3 + 2 = 1331 + 2 = 1333
$$
So, the remainder when $ (2x + 1)^3 + 2 $ is divided by $ x - 5 $ is $ 1333 $.
Conclusion: A Smart Approach with the Remainder Theorem
The Remainder Theorem provides an elegant way to compute remainders without performing full polynomial division. By evaluating $ A(5) $, we simplify $ A(x)^3 + 2 $ to $ A(5)^3 + 2 $, making remainder computation fast and reliable.
