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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
To check polynomial division, multiply the divisor by the quotient and add the remainder. If the resulting polynomial is the same as the original dividend then the division is correct. The justification for this comes straight from the division algorithm for polynomials. In the given example, the calculated resultant polynomial isn't matching the original one, indicating a mistake in our quotient or remainder.

Step by step solution

01

Understanding Polynomial Division

Polynomial division involves dividing a polynomial by another polynomial. This can be checked by multiplying the divisor by the quotient and adding the remainder. If the result equals the original dividend, it confirms the division is correct.
02

Example

Let's take an example where we are given a polynomial \( P(x) = 2x^3 - 3x^2 + 4x - 5 \) and it's divided by \( D(x) = x - 1 \) with quotient \( Q(x) = 2x^2 - x + 3 \) and remainder \( R(x) = 2 \). We can multiply \( D(x) \) by \( Q(x) \) and add \( R(x) \). The result should equal \( P(x) \). i.e., If \( D(x) \cdot Q(x) + R(x) = P(x) \), then polynomial division is correct.
03

Implement Check

So, on implementing this check, we get: \( (x - 1) \cdot (2x^2 - x + 3) + 2 = 2x^3 - x^2 + 3x - 2x^2 + x - 3 + 2 = 2x^3 - 3x^2 + 4x - 1 \neq P(x) \). The results aren't matching, implying there might be a mistake in our quotient and remainder.

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Most popular questions from this chapter

In Exercises 51 - 54, write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \( f(x) = x^4 - 3x^3 - x^2 - 12x - 20 \) (Hint: One factor is \( x^2 + 4 \).)

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