Problem 7 - A Graphical Approach to College Algebra

Question

Each expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{8 x^{3}-5 x}{2 x}$$

Step-by-Step Solution

Answer

The simplified expression is \( 4x^2 - \frac{5}{2} \).

Step 1: Identify the Numerator and the Denominator

The expression given to us is \( \frac{8x^3 - 5x}{2x} \). Here, \(8x^3 - 5x\) is the numerator and \(2x\) is the denominator.

Step 2: Apply the Property Separately

According to the property \( \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} \), we need to split the numerator into separate fractions: \( \frac{8x^3}{2x} + \frac{-5x}{2x} \).

Step 3: Simplify Each Fraction

We simplify each fraction separately:- For the first term, \( \frac{8x^3}{2x} \), we divide the coefficients \( 8 \) by \( 2 \) to get \( 4 \), and simplify \( x^3 \) by \( x \) to get \( x^2 \), resulting in \( 4x^2 \).- For the second term, \( \frac{-5x}{2x} \), the \( x \) terms cancel, leaving us with \( -\frac{5}{2} \).

Step 4: Combine Simplified Terms

Combine the simplified fractions to get the final result: \( 4x^2 - \frac{5}{2} \).

Key Concepts

Rational Expressions

Understanding rational expressions is essential in algebra as these are quite common in many mathematical problems. A rational expression is a fraction where both the numerator and the denominator are polynomials.
In the expression \( \frac{8x^3 - 5x}{2x} \), the numerator is the polynomial \(8x^3 - 5x\), and the denominator is \(2x\). When dealing with rational expressions, the main goal is often to simplify them or perform operations such as addition, subtraction, multiplication, and division.
It's important to ensure that the denominator is not zero, as division by zero is undefined in mathematics. In this expression, \(2x\) should not be equal to zero, which implies that \(x\) should not be zero. Understanding these core aspects helps in making the right decisions while handling such expressions.

Simplification

The process of simplification in algebra involves reducing expressions in their simplest form. It helps in making equations easier to work with and understand.
In rational expressions, like \( \frac{8x^3 - 5x}{2x} \), simplification can often involve separating the terms in the numerator and simplifying each term independently. By applying the property \( \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} \), the numerator can be divided into individual terms.

• The first term \( \frac{8x^3}{2x} \) simplifies by dividing both the coefficient 8 by 2, giving 4, and simplifying \(x^3\) by \(x\), resulting in \(4x^2\).
• The second term \( \frac{-5x}{2x} \) simplifies by cancelling the \(x\) terms, leaving \(-\frac{5}{2}\).

This step-by-step simplification leads to the more manageable expression \(4x^2 - \frac{5}{2}\), which is easier to interpret and use in further calculations.

Polynomial Division

Polynomial division in rational expressions is essential to simplifying and understanding fractions that involve polynomials. In simplified form, polynomial division involves dividing terms separately, as shown in the expression \( \frac{8x^3 - 5x}{2x} \).
Each term of the numerator is divided by the denominator individually. The process of polynomial division here involves:

• Dividing the coefficients of corresponding terms, like \( \frac{8}{2}\) which results in 4.
• Reducing the variable powers, as in \( \frac{x^3}{x} \), which simplifies to \(x^2\).

This results in simplified expressions that are easier to work with, making polynomial division a powerful tool for handling algebraic fractions. Understanding how to properly divide these terms ensures that you can simplify expressions accurately and efficiently.