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. $$ \frac{6 x^{3}-5 x^{2}-7 x-3}{2 x-5} $$

Short answer

The expression is improper. It can be rewritten as \(3x^2 + 5x + 9 + \frac{42}{2x - 5} \).

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 is improper if the degree of the numerator is greater than or equal to the degree of the denominator.

Compare Degrees of Numerator and Denominator

The degree of the numerator, \(6x^{3}-5x^{2}-7x-3\), is 3. The degree of the denominator, \(2x-5\), is 1. Since 3 > 1, the given expression is improper.

Use Polynomial Long Division

Divide the numerator \(6x^{3}-5x^{2}-7x-3\) by the denominator \(2x-5\).

Perform the Division

1. Divide the leading term of the numerator by the leading term of the denominator: \( \frac{6x^{3}}{2x} = 3x^{2} \). 2. Multiply \(3x^{2} \) by \( 2x-5 \) to get \( 6x^{3} - 15x^{2} \). 3. Subtract \( 6x^{3} - 15x^{2} \) from \( 6x^{3}-5x^{2}-7x-3 \) which results in \( 10x^{2} - 7x - 3 \). 4. Repeat this process with the next term: \( \frac{10x^{2}}{2x} = 5x \). Multiply and subtract until the remainder's degree is less than the denominator's degree.

Complete the Division

Continuing, \( \frac{10x^{2}}{2x} = 5x \), multiply and subtract, obtaining \( 5x \times (2x - 5) = 10x^{2} - 25x \), then \( 10x^{2} - 7x - 3 - (10x^{2} - 25x) = 18x - 3 \). Next, \( \frac{18x}{2x} = 9 \), multiply and subtract \( 9 \times (2x - 5) = 18x - 45 \), which yields \( 18x - 3 - (18x - 45) = 42 \). The quotient is \( 3x^{2} + 5x + 9 \) and the remainder is \( 42 \).

Rewrite as a Sum

The improper rational expression \( \frac{6x^{3}-5x^{2}-7x-3}{2x-5} \) can be rewritten as the sum of the polynomial and the proper rational expression: \(3x^2 + 5x + 9 + \frac{42}{2x - 5} \).

Key concepts

proper and improper rational expressions

To fully understand rational expressions, you need to know about proper and improper rational expressions. A rational expression is basically a fraction where the numerator and the denominator are polynomials. But when do we call it proper or improper?

explaining proper and improper

A rational expression is **proper** if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator. For example, in \( \frac{3x^2 + x + 1}{x^3 + 2x + 1} \), the degree of the numerator is 2 and the degree of the denominator is 3, so it's a proper rational expression.

A rational expression is **improper** if the degree of the numerator is greater than or equal to the degree of the denominator. For instance, for \( \frac{6x^3 - 5x^2 - 7x - 3}{2x - 5} \), the degree of the numerator is 3 and the degree of the denominator is 1, making it improper.

polynomial long division

When dealing with improper rational expressions, one useful technique is polynomial long division. This technique helps in rewriting the expression by dividing it into simpler parts.

Think of it like regular long division but with polynomials. Here's a simple guide:

• Write down the numerator and denominator.
• Divide the first term of the numerator by the first term of the denominator.
• Multiply the entire denominator by this result.
• Subtract this product from the numerator.
• Repeat the steps with the new polynomial formed after subtraction until the degree of the remainder is less than the denominator.

For example, when dividing \(6x^{3}-5x^{2}-7x-3\) by \(2x-5\), you perform a series of divisions, multiplications, and subtractions until you can't go further.

degree of polynomials

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial \(4x^3 + 2x^2 + 6x + 1\), the highest power of x is 3, so the degree is 3.

Understanding the degree of polynomials is crucial when determining whether a rational expression is proper or improper.
Always look for the term with the highest exponent to find the degree.
Knowing the degrees helps in operations like polynomial long division and in simplifying rational expressions.

simplifying rational expressions

Simplifying rational expressions means reducing them to their simplest form. This often involves factoring both the numerator and the denominator and then canceling out common factors.

Let's say we have \( \frac{6x^3 - 5x^2 - 7x - 3}{2x - 5} \), and we've already determined it's improper. Using polynomial long division, we separated it into a polynomial and a proper rational expression.
The result was \(3x^2 + 5x + 9 + \frac{42}{2x - 5}\).
Now, we have two parts: the polynomial \(3x^2 + 5x + 9\) and the proper rational expression \(\frac{42}{2x - 5}\), which cannot be simplified further.

Simplifying rational expressions makes them much easier to work with in further calculations or solving equations.