Question

Determine whether the given rational expression is proper or improper. If the expression is improper, rewrite it as the sum of a polynomial and a proper rational expression. $$ \frac{2 x\left(x^{2}+4\right)}{x^{2}+1} $$

Short answer

The rational expression is improper and can be rewritten as \(2x + \frac{6x}{x^2 + 1}\).

Step-by-step solution

- Simplify the Rational Expression

First, simplify the given rational expression: \[ \frac{2x(x^2 + 4)}{x^2 + 1} \].Multiply out the numerator: \[ 2x(x^2 + 4) = 2x^3 + 8x. \]The simplified expression is: \[ \frac{2x^3 + 8x}{x^2 + 1}. \]

- Determine Proper or Improper

Compare the degrees of the numerator and the denominator.The degree of the numerator \(2x^3 + 8x\) is 3 and the degree of the denominator \(x^2 + 1\) is 2.Since the degree of the numerator (3) is greater than the degree of the denominator (2), the rational expression is improper.

- Perform Polynomial Division

To rewrite the improper rational expression as the sum of a polynomial and a proper rational expression, perform polynomial division.Divide \(2x^3 + 8x\) by \(x^2 + 1\):1. Divide the leading term: \ \(\frac{2x^3}{x^2} = 2x\)2. Multiply: \(2x \cdot (x^2 + 1) = 2x^3 + 2x\)3. Subtract: \(2x^3 + 8x - (2x^3 + 2x) = 6x\)4. The quotient is \(2x\) and the remainder is \(6x\).

- Write the Final Expression

Combine the quotient and the remainder:\[ 2x + \frac{6x}{x^2 + 1}. \]The rational expression can thus be rewritten as the sum of the polynomial \(2x\) and the proper rational expression \(\frac{6x}{x^2 + 1}\).

Key concepts

Proper Rational Expression

Let's first talk about proper rational expressions. A rational expression is considered **proper** if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator. To better understand this, let's break down each term. The *degree* of a polynomial is the highest power of the variable in the polynomial. For instance, in the polynomial \(2x^3 + 8x\), the degree is 3, because the highest power of \(x\) is 3. Similarly, in \(x^2 + 1\), the degree is 2. So, if you see a rational expression where the numerator's degree is less than the denominator's, it’s a proper rational expression. In simpler terms, if the numerator polynomials don’t overtake the polynomials in the denominator in their 'power' or 'degree', it’s proper.

Improper Rational Expression

Now, let’s look at improper rational expressions. An expression is considered **improper** when the degree of the numerator is equal to or greater than the degree of the denominator. In our given exercise, the rational expression is \(\frac{2x^3 + 8x}{x^2 + 1}\). The degree of the numerator \(2x^3 + 8x\) is 3, while the degree of the denominator \(x^2 + 1\) is 2. Because 3 (numerator degree) is greater than 2 (denominator degree), this expression is improper. The key is recognizing the largest power of the variable: if the numerator's power is at least as great as the denominator's, you have an improper rational expression. Such cases often require rewriting to express them in more straightforward terms.

Polynomial Division

When faced with an improper rational expression, we use **polynomial division** to break it down into a simpler form. Here's how polynomial division works:
1. **Divide the Leading Terms**: First, divide the leading term of the numerator by the leading term of the denominator. In our example, we divide \(2x^3\) by \(x^2\), which gives us \(2x\).
2. **Multiply and Subtract**: Next, multiply the entire denominator by the result of the first step and subtract this product from the numerator. For our case: \(2x \cdot (x^2 + 1) = 2x^3 + 2x\). Subtracting, we get \(2x^3 + 8x - (2x^3 + 2x) = 6x\).
3. **Combine Results**: The quotient from the division (\(2x\)) combined with the remainder (\(6x\)) gives us the polynomial part \(2x\), and the new remainder forms the proper rational expression part \(\frac{6x}{x^2 + 1}\).
Putting these together, the expression rewrites as the sum of the polynomial \(2x\) and the proper rational expression \(\frac{6x}{x^2 + 1}\), making it easier to work with and understand.