Question

Use the division algorithm to rewrite each improper rational expression as the sum of a polynomial and a proper rational expression. Find the partial fraction decomposition of the proper rational expression. Finally, express the improper rational expression as the sum of a polynomial and the partial fraction decomposition. $$ \frac{x^{3}+x}{x^{2}+4} $$

Short answer

Using polynomial long division and partial fractions, \(\frac{x^3 + x}{x^2 + 4} = x + \frac{-3x}{x^2 + 4}\).

Step-by-step solution

Perform Polynomial Long Division

Divide the numerator \(x^3 + x\) by the denominator \(x^2 + 4\). First, divide the leading term of the numerator by the leading term of the denominator: \(\frac{x^3}{x^2} = x\).Next, multiply \(x\) by the entire denominator \(x^2 + 4\), which gives \(x \cdot (x^2 + 4) = x^3 + 4x\).Then, subtract \(x^3 + 4x\) from the original numerator \(x^3 + x\) to find the remainder: \(x^3 + x - (x^3 + 4x) = x - 4x = -3x\).

Write as Sum of Polynomial and Proper Rational Expression

From the division, \(\frac{x^3 + x}{x^2 + 4} = x + \frac{-3x}{x^2 + 4}\). Now, express the result as the sum of the polynomial part and the proper rational expression: \(x + \frac{-3x}{x^2 + 4}\).

Find Partial Fraction Decomposition

Express the proper rational expression \(\frac{-3x}{x^2 + 4}\) in partial fractions. Since the denominator \(x^2 + 4\) cannot be factored further over the real numbers, it is already in its simplest form. As such, no further decomposition is needed, leaving \(\frac{-3x}{x^2 + 4}\) as it is.

Combine Results

Combine the polynomial part and the partial fraction decomposition to express the original improper rational expression. Thus, \(\frac{x^3 + x}{x^2 + 4}\) can be expressed as: \(x + \frac{-3x}{x^2 + 4}\).

Key concepts

division algorithm

The division algorithm is a fundamental technique that connects division with multiplication and addition. This is especially useful for polynomials. For any polynomials, say, the dividend \(f(x)\) and the divisor \(g(x)\) with \(deg(f(x)) \geq deg(g(x))\), there exist unique polynomials \(q(x)\) (quotient) and \(r(x)\) (remainder) such that:
division algorithm formula: \[f(x) = g(x)q(x) + r(x), where \ deg(r(x)) < deg(g(x))\]
This formula ensures the division of polynomials until the degree of the remainder is less than the degree of the divisor. When applied to improper rational expressions, like \[\frac{x^3 + x}{x^2 + 4}\],the goal is to break it down into a polynomial plus a proper fraction. This makes complex functions easier to work with and understand.

polynomial long division

Polynomial long division is similar to numerical long division, but it operates with polynomials. Let's break down the steps using the example \[\frac{x^3 + x}{x^2 + 4}. \]1. **Divide leading terms**: Divide the leading term of the numerator \(x^3\) by the leading term of the denominator \(x^2\): \[\frac{x^3}{x^2} = x\].
2. **Multiply and subtract**: Multiply \(x\) by the entire divisor \(x^2 + 4\), giving \[x \times (x^2 + 4) = x^3 + 4x\]. Then, subtract this result from the original polynomial: \[x^3 + x - (x^3 + 4x) = -3x\].

3. **Continue until no more division**: Here, \(-3x\) is the remainder, which cannot be divided further by \(x^2 + 4\) because its degree is less than that of the divisor. Thus, the expression can be rewritten as: \[x + \frac{-3x}{x^2 + 4}\].

partial fraction decomposition

Partial fraction decomposition is a method used to break down a complex rational expression into a sum of simpler fractions. This tool is particularly useful when integrating functions or when solving differential equations. In our example, we need to decompose the proper rational expression obtained from the division step. The proper fraction part here is: \[\frac{-3x}{x^2 + 4}\].
Since \(x^2 + 4\) cannot be factored further over the real numbers, it remains as it is. Thus, the partial fraction decomposition does not need further steps here. While more complex examples may require breaking down further, this particular fraction remains \[\frac{-3x}{x^2 + 4}\] in its simplest form.

proper rational expression

A rational expression is called proper if the degree of the numerator is less than the degree of the denominator. Proper rational expressions are much simpler to work with than improper ones. In our case, after performing polynomial long division on the improper rational expression, we obtained:\[\frac{x^3 + x}{x^2 + 4} = x + \frac{-3x}{x^2 + 4}\].
The \(\frac{-3x}{x^2 + 4}\) term is a proper rational expression as the degree of \(-3x\) (which is 1) is less than the degree of \(x^2 + 4\) (which is 2). This step not only simplifies the expression but also makes it easier to apply further mathematical operations like integration or differentiation.
By understanding and converting improper rational expressions into sums involving polynomials and proper rational expressions, we can simplify complex algebraic manipulations and enhance our problem-solving toolbox. This approach is particularly valuable in calculus and higher-level mathematics.