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. $$ \frac{5 x^{3}+2 x-1}{x^{2}-4} $$

Short answer

Improper; rewritten as \( 5x + \frac{22x - 1}{x^2 - 4} \)

Step-by-step solution

Identify Degrees of the Numerator and Denominator

Determine the degrees of the numerator and the denominator. In the given expression, the numerator is \(5x^3 + 2x - 1\). The degree of the numerator is 3 because the highest power of \(x\) is 3. The denominator is \(x^2 - 4\). The degree of the denominator is 2 because the highest power of \(x\) is 2.

Determine if the Expression is Proper or Improper

Compare the degrees of the numerator and denominator. Since the degree of the numerator (3) is greater than the degree of the denominator (2), the rational expression \(\frac{5x^3 + 2x - 1}{x^2 - 4}\) is improper.

Perform Polynomial Long Division

Divide the numerator \(5x^3 + 2x - 1\) by the denominator \(x^2 - 4\). Perform the polynomial long division to obtain the quotient and remainder.

Quotient and Remainder

The result of the polynomial long division is a quotient of \(5x\) and a remainder of \(22x - 1\).

Write as Sum of Polynomial and Proper Rational Expression

Express the original improper rational expression as the sum of the polynomial quotient and a proper rational expression. The final answer is \[ 5x + \frac{22x - 1}{x^2 - 4} \]

Key concepts

Proper and Improper Fractions

Understanding whether a rational expression is proper or improper is crucial in algebra. A rational expression is the fraction form of two polynomials, where the numerator (top part) and the denominator (bottom part) are both polynomials.

When determining if a rational expression is proper or improper, compare the degrees of the numerator and the denominator:

• A proper fraction has a numerator with a lower degree than the denominator.
• An improper fraction has a numerator with a degree equal to or greater than the degree of the denominator.

In the given example of \(\frac{5x^3 + 2x - 1}{x^2 - 4}\), the degree of the numerator (3) is greater than the degree of the denominator (2), making it an improper fraction. Once identified as improper, you can often simplify these expressions using polynomial long division.

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to the long division of numbers. It helps break down an improper fraction into a simpler form.

Here’s a simplified process for performing polynomial long division:

• Divide the leading term of the numerator by the leading term of the denominator. In our example, divide \(5x^3\) by \(x^2\) to get \(5x\).
• Multiply the entire divisor (denominator) by the result from the previous step. Multiply \(5x\) by \(x^2 - 4\) to get \(5x^3 - 20x\).
• Subtract this result from the original numerator. Subtract \(5x^3 - 20x\) from \(5x^3 + 2x - 1\) to get \(22x - 1\).
• Repeat the steps using the new polynomial until the degree of the remainder is less than the degree of the original divisor.

In our problem, after dividing \(5x^3 + 2x - 1\) by \(x^2 - 4\), the quotient is \(5x\), and the remainder is \(22x - 1\). The original improper fraction can thus be rewritten as the sum of the polynomial quotient \(5x\) and a proper rational expression \(\frac{22x - 1}{x^2 - 4}\).

Degree of a Polynomial

The degree of a polynomial is a crucial concept in understanding rational expressions and polynomial division. It is determined by the highest power of the variable in the polynomial.

For example:

• In \(5x^3 + 2x - 1\), the highest power of \(x\) is 3, so the degree is 3.
• In \(x^2 - 4\), the highest power of \(x\) is 2, so the degree is 2.

Knowing the degrees helps identify proper and improper fractions and is essential for performing polynomial long division. When comparing the degrees:

• If the degree of the numerator is lower, the fraction is proper.
• If the degree of the numerator is equal to or greater than the denominator's degree, the fraction is improper, and polynomial long division can be used to simplify it.

The concept of polynomial degree underpins many algebraic operations and helps in the simplification of complex rational expressions.