Problem 45 - Algebra and Trigonometry

Question

perform each long division and write the partial fraction decomposition of the remainder term. $$\frac{x^{4}-x^{2}+2}{x^{3}-x^{2}}$$

Step-by-Step Solution

Answer

The result of the long division is \(x + 1\), and the partial fraction decomposition of the remainder term is \(\frac{A}{x^{2}} + \frac{B}{x - 1}\), where the actual values of A and B depend on solving the resulting equation from the comparison of coefficients. Hence, the original fraction can be expressed as \(x + 1\) plus the aforementioned partial fraction decomposition.

Step 1: Polynomial Division

Perform the polynomial division of \(x^{4}-x^{2}+2\) by \(x^{3}-x^{2}\). \nTo do this, we first divide the leading term in the numerator, \(x^4\), by the leading term in the denominator, \(x^3\), to get \(x\). This is the first term of our quotient. \nMultiplying \(x^3 - x^2\) by \(x\) gives \(x^4 - x^3\). Subtracting this from our original numerator gives a new numerator of \(x^3 + x^2 + 2\). \nWe repeat the process for the new numerator. The leading term, \(x^3\), divided by \(x^3\) gives a \(1\). Subtracting \(x^3 - x^2\) multiplied by \(1\) gives us \(2x^2 + 2\). \nSince the degree of the new numerator \(2x^2 + 2\) is less than the degree of the denominator, we stop here. Thus, the quotient is \(x + 1\) and the remainder is \(2x^2 + 2\).

Step 2: Express the Division as a Sum

Express the original fraction as a sum of the quotient and the remainder divided by the original denominator. Here, we have \(\frac{x^{4}-x^{2}+2}{x^{3}-x^{2}} = x + 1 + \frac{2x^2 + 2}{x^{3} - x^{2}}\).

Step 3: Partial Fraction Decomposition

Write the remainder term, \(\frac{2x^2 + 2}{x^{3} - x^{2}}\), as a partial fraction. \nThe denominator factors as \(x^{2}(x - 1)\), so we express the fraction as \(\frac{A}{x^{2}} + \frac{B}{x - 1}\), for some constants A and B. Multiplying through by \(x^{2}(x - 1)\) and comparing coefficients on both sides of the equation will give the values of A and B. \nAs a result, we get the partial fraction decomposition of the original fraction.

Key Concepts

Polynomial Division

When dealing with algebraic expressions, particularly polynomials, it's essential to understand how to divide them, much like you would with numbers. Polynomial division is the process of dividing a polynomial by another polynomial, similar to long division in arithmetic.

Let's look at the polynomial division of \(x^{4}-x^{2}+2\) by \(x^{3}-x^{2}\). The goal is to divide the terms of the numerator by those of the denominator to find a quotient and possibly a remainder. We begin by examining the lead terms, dividing \(x^4\) by \(x^3\), which yields \(x\). This first step is crucial as it sets the pace for the rest, as we subtract and bring down terms, similar to traditional long division.

Each term in the quotient represents a part of the division, and when you cannot divide further due to the numerator's degree being lower than the denominator's, you've found your remainder. Understanding this fundamental process is a stepping stone towards mastering more complex algebraic manipulations.

Long Division of Polynomials

Long division of polynomials is an extension of the familiar long division process used with numbers. The method systematically divides a polynomial by another, subsequent subtraction of products, and brings down the next term just as you learned in elementary school. Here’s a brief overview:

• Write both polynomials in descending order of powers.
• Divide the first term of the numerator by the first term of the denominator to get the first term of the quotient.
• Multiply the entire divisor by this new term and subtract from the numerator.
• Bear in mind that the degree of the remainder must always be less than the degree of the divisor. If it’s not, repeat the process till it is.

While doing long division of polynomials, remember to align like terms vertically during each step to avoid confusion. When properly understood and executed, this technique simplifies the understanding of higher-order polynomial behavior in various mathematical contexts.

Remainder Theorem

The Remainder Theorem is an insightful result that deals with the division of polynomials. It states that if a polynomial \(f(x)\) is divided by a binomial of the form \((x - c)\), the remainder is the same as the value you get when you plug \(c\) into the polynomial \(f(x)\).

This theorem has profound implications; for example, it implies that if the remainder is 0, \(c\) is a root of the polynomial. When approached through polynomial division, if the result is not an exact division (i.e., no remainder), the theorem assists in determining the value of that remainder simply by evaluating the polynomial at a specific point.

Expanding the Remainder Theorem, we realize how crucial understanding the division of polynomials is. It's not just about finding the quotient; it's also about gleaning deeper insights into the behavior of polynomials, which is invaluable in fields ranging from algebra to calculus.