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Chapter 2: Problem 94

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

Short Answer

Expert verified
Use the equation \(a = b \cdot q + r\) to validate the polynomial division, whereby \(a\) is original polynomial, \(b\) is divisor, \(q\) is quotient and \(r\) is remainder. Multiply \(b\) and \(q\) and add \(r\) to the product, the result should equal the original polynomial, \(a\). For the example, \((x^3 - 3x^2 + 7x - 6) =(x - 2) \cdot (x^2 - x + 3) + 0\) we can confirm that the division is correct.

Step by step solution

01

Performing Polynomial Division

Do polynomial division like normal long division, but for polynomials. As an example, consider \((x^3 - 3x^2 + 7x - 6) 梅 (x - 2)\). The quotient obtained is \(x^2 - x + 3\) and the remainder is 0.
02

Validating the Division

To verify the division, multiply the divisor, quotient, and then add the remainder. In the example provided, because the remainder is 0 and the divisor is \(x - 2\) and the quotient is \(x^2 - x + 3\). The multiplication works out to be the original polynomial, \(x^3 - 3x^2 + 7x - 6\). Setting equal to the original polynomial allows us to check the division.
03

Confirming the Result

Our original polynomial and our resultant polynomial from validation are the same, thus affirming our solution.

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