Problem 43 - College Algebra Essentials

Question

perform each long division and write the partial fraction decomposition of the remainder term. $$ \frac{x^{5}+2}{x^{2}-1} $$

Step-by-Step Solution

Answer

The partial fraction decomposition of given polynomial is: \(x^{3}- \frac{x^{2}+x-1}{x-1} + \frac{x^{2}-x-1}{x+1}\)

Step 1: Polynomial Division

Firstly, perform polynomial division. Divide \(x^{5}+2\) by \(x^{2}-1\). This can be done using long division or synthetic division. The quotient will be \(x^{3}\) and the remainder will be \(x^{3}+2\). So the original expression can be rewritten as: \(x^{3}+ \frac{x^{3}+2}{x^{2}-1}\).

Step 2: Factorize the denominator

The denominator in the second term, \(x^{2}-1\), can be factorized as \((x-1)(x+1)\). Replace this in the previous expression to get: \(x^{3}+ \frac{x^{3}+2}{(x-1)(x+1)}\).

Step 3: Partial Fraction Decomposition

Now, decompose the fraction \(\frac{x^{3}+2}{(x-1)(x+1)}\) as a sum of two fractions \(\frac{A}{x-1}\) and \(\frac{B}{x+1}\), where A and B are constants to figure out. To determine A and B, equalize the expression \(x^{3}+2=A(x+1)+B(x-1)\) with the same power of x on both sides, solve the system of equations and get \(A=-x^{2}-x+1\) and \(B=x^{2}-x-1\). The original expression can then be decomposed into: \(x^{3}- \frac{x^{2}+x-1}{x-1} + \frac{x^{2}-x-1}{x+1}\).

Key Concepts

Polynomial Long Division

Polynomial long division is a technique used in algebra to divide one polynomial by another, larger, polynomial. It's analogous to the long division process with numbers that many students learn in elementary school.

The process involves writing down the dividend (in our example, \(x^{5}+2\)) and the divisor (\(x^{2}-1\)), then determining how many times the divisor fits into the dividend. Just like with numerical long division, we subtract the result of this multiplication from the dividend and bring down the next term in the polynomial, repeating the process until the degrees of the remaining polynomial are less than the degree of the divisor, producing a remainder. In our exercise, the quotient is \(x^{3}\) and the remainder is \(x^{3}+2\).

The outcome of polynomial long division is incredibly useful because it helps to simplify algebraic expressions that might otherwise be too complex to handle. It breaks down intimidating polynomials into more manageable pieces, paving the way for further algebraic manipulation like factorization or partial fraction decomposition.

Factorization

Factorization is a fundamental process in algebra where we express a polynomial as a product of its factors. Factors are simpler polynomials whose multiplied value equals the original polynomial. This technique is crucial when we're dealing with polynomial expressions, especially when they form the denominator in a fraction, as factorization can often reveal insights into the behavior of the algebraic expression that were not immediately evident.

In the case of our example, \(x^{2}-1\), this is a difference of squares and can be factored into \((x-1)(x+1)\). Recognizing patterns like the difference of squares is a key skill in successfully factorizing expressions, as this insight allows us to decompose what might initially seem like a single inseparable polynomial into a product of binomials. Through factorization, we can simplify complex problems and set the stage for partial fraction decomposition when dealing with rational expressions.

Algebraic Expressions

Algebraic expressions represent mathematical relationships using numbers, variables, and arithmetic operations. They can be simple, with just one variable, or complex, involving multiple terms and powers, as seen in our polynomial \(x^{5}+2\).

The manipulation of these expressions, through operations like addition, subtraction, multiplication, division, and exponentiation, is at the heart of algebra. Our aim is to often simplify these expressions or solve equations derived from them. To work with algebraic expressions effectively, understanding their structure and the underlying principles—such as the distributive property, combining like terms, and using inverse operations—is essential.

In our problem, after performing polynomial long division and factorizing the denominator, we tackle the algebraic expression through partial fraction decomposition. This involves expressing the single complex fraction as a sum of simpler fractions, facilitating further analysis and integration of the expression if needed. Such methods demonstrate the power and flexibility of algebra when dissecting and understanding mathematical relationships.