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. $$ \frac{3 x^{4}+x^{2}-2}{x^{3}+8} $$

Short answer

The expression \( \frac{3x^4 + x^2 - 2}{x^3 + 8} \) is improper. Rewritten it is: \[ 3x + \frac{x^2 - 24x - 2}{x^3 + 8} \].

Step-by-step solution

- Identify Degrees of the Polynomial

The degrees of the polynomial in the numerator and the denominator need to be identified. The degree of the numerator is the highest power of the variable in the numerator, and the degree of the denominator is the highest power of the variable in the denominator. Here, the numerator is \(3x^4 + x^2 - 2\) (degree 4) and the denominator is \(x^3 + 8\) (degree 3).

- Determine Properness

A rational expression is proper if the degree of the numerator is less than the degree of the denominator. Here, the degree of the numerator (4) is greater than the degree of the denominator (3), so the given expression is improper.

- Perform Polynomial Division

To rewrite the improper fraction as the sum of a polynomial and a proper rational expression, perform polynomial division. Divide \(3x^4 + x^2 - 2\) by \(x^3 + 8\).

- Divide the Leading Terms

Divide the leading term of the numerator by the leading term of the denominator: \( \frac{3x^4}{x^3} = 3x \). This is the first term of the quotient.

- Multiply and Subtract

Multiply \(3x\) by \(x^3 + 8\) yielding \(3x^4 + 24x\). Subtract this from the original numerator: \( (3x^4 + x^2 - 2) - (3x^4 + 24x) = x^2 - 24x - 2 \).

- Continue Division

Next, divide the new leading term of the remainder by the leading term of the denominator: \(\frac{x^2}{x^3}\). Since the degree of the remaining term is now less than the degree of the denominator, the division stops here.

- Rewrite the Expression

Combine the results to rewrite the improper fraction as the sum of the polynomial and the proper rational expression. The result is: \[ 3x + \frac{x^2 - 24x - 2}{x^3 + 8} \]

Key concepts

Proper Rational Expression

When working with rational expressions, it's important to determine whether an expression is proper or not. A rational expression is made up of a numerator and a denominator, both of which are polynomials. The key thing you need to look at is the degree of each polynomial.
The degree of a polynomial is the highest power of the variable that appears in it. For example, for the polynomial \(3x^4 + x^2 - 2\), the highest power is 4, so its degree is 4. Now, if the degree of the numerator (the polynomial on top) is less than the degree of the denominator (the polynomial on the bottom), the rational expression is called proper.
In our specific example here, we have the fraction \(\frac{3x^4 + x^2 - 2}{x^3 + 8}\). The degree of the numerator is 4, and the degree of the denominator is 3. Since the degree of the numerator is greater than that of the denominator, this rational expression is not proper; it is improper.

Degree of Polynomial

Understanding the 'degree of a polynomial' is crucial for working with rational expressions. To find the degree, simply identify the term with the highest exponent. This highest exponent determines the polynomial's degree.
In a polynomial like \(3x^4 + x^2 - 2\), the highest exponent is 4. So, the degree is 4. Similarly, for \(x^3 + 8\), the highest exponent is 3, making the degree 3. Knowing these degrees helps us determine if a rational expression is proper or improper.
When the numerator's degree is lower than the denominator's degree, the rational expression is proper. If the numerator's degree is higher (or even equal), the expression is improper. In this context, for \(\frac{3x^4 + x^2 - 2}{x^3 + 8}\), because 4 (numerator) > 3 (denominator), the rational expression is improper.

Polynomial Division

To handle an improper rational expression, polynomials can be divided similar to numbers. The aim is to express the improper fraction as a polynomial plus a proper rational expression.
The steps include:

• Dividing the leading term of the numerator by the leading term of the denominator.
• Using the quotient to multiply the entire denominator, then subtract this result from the numerator.
• Repeating until the remainder's degree is less than the denominator's degree.

For example, in the expression \(\frac{3x^4 + x^2 - 2}{x^3 + 8}\), we divide \(3 x^4\) by \(x^3\), getting \(3x\).
Multiplying \(3x\) with \(x^3 + 8\) gives us \(3x^4 + 24x\).
After subtracting this from our original numerator, what remains is \(x^2 - 24x - 2\).
We stop here because the degree of \(x^2 - 24x - 2\) (which is 2) is less than the degree of \(x^3 + 8\) (which is 3). Therefore, we can rewrite \(\frac{3x^4 + x^2 - 2}{x^3 + 8}\) as \(3x + \frac{x^2 - 24x - 2}{x^3 + 8}\).
This finally gives us a proper rational expression.