Problem 80 - College Algebra and Calculus: An Applied Approach

Question

Writing Briefly explain how to check polynomial division, and justify your answer. Give an example.

Step-by-Step Solution

Answer

To check polynomial division, you multiply the divisor by the quotient, then add the remainder. The result should be equivalent to the original dividend. This verifies that the division was performed correctly.

Step 1: Understanding Polynomial Division

Polynomial division works the same way as numerical division that you learned in elementary school. The only difference is that instead of numbers, you have expressions. In the division process, you are given a dividend (the polynomial you are dividing) and a divisor (the polynomial by which you are dividing). The result of the division is the quotient and there may also be a remainder, which is a polynomial with a degree less than the divisor.

Step 2: Procedure to Verify Polynomial Division

To verify the result of a polynomial division, multiply the divisor by the quotient, then add the remainder. The result should be equivalent to the original dividend. This proves that the division was performed correctly.

Step 3: Example of Verifying Polynomial Division

For instance, say you divide \(x^3 - 3x^2 + 5x + 1\) by \(x - 2\) and get a quotient of \(x^2 - x - 3\) with a remainder of \(7\). To check this, you multiply \(x - 2\) by \(x^2 - x - 3\), which results in \(x^3 - 3x^2 + 5x - 6\). Adding the remainder of \(7\) gives you \(x^3 - 3x^2 + 5x + 1\), which is the original dividend, thereby confirming the correctness of the division.

Key Concepts

Verifying Polynomial Division

When you've divided two polynomials, it's crucial to ensure that the operation was carried out correctly. Verifying the result of polynomial division is quite straightforward and involves a simple check. Begin by multiplying the calculated quotient by the divisor. Then, add the remainder to this product. If the division is correct, this sum should match the original dividend exactly.

To illustrate, let's assume you've been given the task of dividing the polynomial \(x^3 - 3x^2 + 5x + 1\) by \(x - 2\) and you've determined that the quotient is \(x^2 - x - 3\) with a remainder of 7. To verify this result, perform the multiplication \((x - 2)(x^2 - x - 3)\), which simplifies to \(x^3 - 3x^2 + 5x - 6\). Now add the remainder, yielding \(x^3 - 3x^2 + 5x - 6 + 7\), simplifying to the original dividend \(x^3 - 3x^2 + 5x + 1\). If the sum matches the original dividend, the division process was correct. This verification step is essential to ensure accuracy in your polynomial division operations.

Division of Polynomials

Understanding how to divide polynomials is a fundamental skill in algebra. Much like the division of numbers, polynomial division involves finding how many times the divisor can fit into the dividend. The key parts to remember are the dividend (the polynomial being divided), the divisor (the polynomial to divide by), the quotient (the result of the division), and often a remainder (what's left over, if anything).

The process often utilizes long division or synthetic division techniques. For example, when dividing \(x^3 - 3x^2 + 5x + 1\) by \(x - 2\), you work through each term systematically. The key steps involve determining how many times the leading term of the divisor fits into the leading term of the dividend, performing the multiplication, subtracting this from the dividend, and continuing the process with the new, reduced polynomial dividend. If there's a remainder, it will be a polynomial of a lower degree than the divisor or a constant. Practicing this method with different polynomials enhances understanding and technique proficiency.

Mathematical Proof

A mathematical proof is an evidence-based method used to ascertain the truth of a statement. In essence, proofs provide a logical and sequential argument to demonstrate why a mathematical concept conforms to established principles and axioms. For polynomial division, the proof lies in reversing the process: by taking the quotient and multiplying it by the divisor, then adding any remainder, we should recover the original polynomial (dividend).

If this condition is satisfied, we've effectively proven that our division was correct. The necessity of a valid proof in mathematics stems from the need for certainty and clarity; it is not enough to simply state a result—we must also provide a convincing argument that establishes its legitimacy. Proofs can vary in complexity, from simple arithmetic checks to more elaborate logical arguments, depending on the mathematical topic being explored.