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. $$ \frac{5 x+2}{x^{3}-1} $$

Short answer

The rational expression \frac{5x + 2}{x^{3} - 1} is proper.

Step-by-step solution

- Compare the degrees

Check the degrees of the numerator and the denominator. The degree of the numerator (5x + 2) is 1, and the degree of the denominator (x^3 - 1) is 3.

- Determine if the expression is proper or improper

A rational expression is considered improper if the degree of the numerator is greater than or equal to the degree of the denominator. Since the degree of the numerator (1) is less than the degree of the denominator (3), the expression is proper.

Key concepts

proper rational expression

A proper rational expression is a type of fraction where the degree of the numerator is less than the degree of the denominator. For instance, in the expression \(\frac{5x+2}{x^{3}-1}\), the numerator is \(5x+2\) and the denominator is \(x^{3}-1\). The degree of the numerator is 1 (because of the term \(5x\)) and the degree of the denominator is 3 (because of the term \(x^3\)). Since 1 < 3, this is a proper rational expression.

• A rational expression is proper if the highest power of x in the numerator is less than the highest power of x in the denominator.
• This means the numerator 'fits' nicely under the denominator, like a bounded fraction.

Understanding whether a rational expression is proper or not can influence how you simplify or manipulate it in algebra. Knowing that the expression \(\frac{5x+2}{x^{3}-1}\) is proper tells us that the fraction is already in its simplest form regarding the relationship between the numerator and denominator.

improper rational expression

An improper rational expression is one where the degree of the numerator is greater than or equal to the degree of the denominator. If we look at an expression like \(\frac{x^3+2x^2+x}{x^2-1}\), both the degrees for the numerator and the denominator are equal to 3; therefore, it is an improper rational expression.

• A rational expression is improper if the highest power of x in the numerator is greater than or equal to the highest power of x in the denominator.
• To simplify such expressions, we can use polynomial long division, which rewrites the improper rational expression as the sum of a polynomial and a proper rational expression.

Transforming an improper rational expression often involves separating out the polynomial part and the remaining proper fraction. This can make integration or further algebraic manipulation simpler.

degree of polynomial

The degree of a polynomial refers to the highest power of the variable within the polynomial. For example, in the polynomial \(5x^2 + 3x + 7\), the degree is 2 because \(x^2\) is the highest power present in the expression.

• Degrees help us compare different polynomials and rational expressions.
• Checking the degree is fundamental in determining if a rational expression is proper or improper.

In our exercise, we looked at \(\frac{5x+2}{x^{3}-1}\). The numerator \(5x + 2\) has a degree of 1, and the denominator \(x^3 - 1\) has a degree of 3. Since 1 is less than 3, we concluded that the expression is proper. Learning to identify and compare degrees will help simplify many algebraic tasks.