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. $$ \frac{x^{3}+12 x^{2}-9 x}{9 x^{2}-x^{4}} $$

Short answer

Proper because degree of the numerator (3) < degree of the denominator (4).

Step-by-step solution

Understand Proper and Improper Rational Expressions

A rational expression is proper if the degree of the numerator is less than the degree of the denominator. It's improper if the degree of the numerator is equal to or greater than the degree of the denominator.

Determine the Degrees

Identify and compare the degrees of the numerator and the denominator. The degree is the highest power of the variable in the polynomial.Numerator: \(x^3 + 12x^2 -9x\) (degree 3)Denominator: \(9x^2 -x^4\) (degree 4)

Compare the Degrees

Compare the degree of the numerator (3) with the degree of the denominator (4). Since 3 < 4, the given rational expression is proper.

Key concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients.
For example, in the rational expression \(\frac{2x^2 + 3x + 1}{x - 4}\), \(2x^2 + 3x + 1\) is the numerator and \(x - 4\) is the denominator.
Understanding rational expressions is fundamental for simplifying, adding, subtracting, multiplying, and dividing expressions.

Degree of Polynomials

The degree of a polynomial is the highest power of the variable in the polynomial. In other words, it tells you the most times the variable is multiplied by itself in the expression.

For example:

• The degree of \(3x^4 + 2x^2 + x + 7\) is 4, since the highest power of \(x\) is \(x^4\).
• The degree of \(x^3 + 12x^2 - 9x\) is 3, since the highest power of \(x\) is \(x^3\).
• The degree of \(9x^2 - x^4\) is 4, since the highest power of \(x\) is \(x^4\).

Knowing the degree of the polynomials in both the numerator and the denominator is crucial in identifying proper and improper rational expressions.

Numerator and Denominator Comparison

To determine whether a rational expression is proper or improper, compare the degrees of the numerator and the denominator.

A rational expression is considered proper if the degree of the numerator is less than the degree of the denominator. If the degree of the numerator is equal to or greater than the degree of the denominator, the rational expression is improper.

Consider the example \(\frac{x^3 + 12x^2 - 9x}{9x^2 - x^4}\):

• The degree of the numerator \(x^3 + 12x^2 - 9x\) is 3.
• The degree of the denominator \(9x^2 - x^4\) is 4.

Since 3 < 4, the rational expression \(\frac{x^3 + 12x^2 - 9x}{9x^2 - x^4}\) is classified as a proper rational expression.
If the comparison results in the numerator's degree being equal to or greater than the denominator's degree, additional steps may be required to rewrite the expression as the sum of a polynomial and a proper rational expression.