Problem 4 - College Algebra with Modeling and Visualization

Question

Divide the expression. $$\frac{5 x^{3}-10 x^{2}+5 x}{15 x^{2}}$$

Step-by-Step Solution

Answer

\( \frac{1}{3}x - \frac{2}{3} + \frac{1}{3x} \)

Step 1: Break Down the Problem

You need to simplify the expression \( \frac{5x^3 - 10x^2 + 5x}{15x^2} \). This involves dividing each term in the numerator by the denominator, which is \( 15x^2 \).

Step 2: Divide Each Term Separately

Divide the first term: \( \frac{5x^3}{15x^2} = \frac{5}{15} \cdot \frac{x^3}{x^2} = \frac{1}{3}x \). Divide the second term: \( \frac{-10x^2}{15x^2} = \frac{-10}{15} \cdot \frac{x^2}{x^2} = -\frac{2}{3} \). Divide the third term: \( \frac{5x}{15x^2} = \frac{5}{15} \cdot \frac{1}{x} = \frac{1}{3x} \).

Step 3: Combine the Simplified Terms

Now, combine all the individual terms obtained: \( \frac{1}{3}x - \frac{2}{3} + \frac{1}{3x} \).

Key Concepts

Polynomial Division

Polynomial division involves dividing one polynomial by another, often simplifying the expression involved. In our example, the division is represented as \( \frac{5x^3 - 10x^2 + 5x}{15x^2} \). This expression consists of a polynomial numerator and a monomial denominator.

The key idea is to divide each term in the numerator individually by the denominator. This method, known as term-by-term division, allows us to manage the complexity step by step.

• Step 1: Identify each term of the numerator. In this case, we have three terms: \( 5x^3 \), \( -10x^2 \), and \( 5x \).
• Step 2: Divide each term by the denominator \( 15x^2 \).
• Calculate: Apply division separately for each term to simplify properly.

Polynomial division simplifies the relationship between the polynomials, providing a more manageable form for further analysis or calculation.

Simplification of Expressions

Simplifying algebraic expressions is about reducing them to their simplest form while maintaining equivalency. The simplification process applied in the example makes the expression more understandable and easier to work with.

To simplify the polynomial division, each term in the expression \( \frac{5x^3 - 10x^2 + 5x}{15x^2} \) was divided by \( 15x^2 \).

• First term: The expression \( \frac{5x^3}{15x^2} \) simplifies to \( \frac{1}{3}x \).
• Second term: The expression \( \frac{-10x^2}{15x^2} \) simplifies to \( -\frac{2}{3} \).
• Third term: The expression \( \frac{5x}{15x^2} \) simplifies to \( \frac{1}{3x} \).

Bringing these together yields the simplified form: \( \frac{1}{3}x - \frac{2}{3} + \frac{1}{3x} \). This simpler expression is easier to interpret and use in future calculations.

Fractional Coefficients

Fractional coefficients emerge when terms within algebraic expressions are divided, particularly in the context of polynomial division. They occur frequently and can initially seem complex, but they simplify expressions by scaling each term to a more manageable size.

The key step in handling fractional coefficients is the reduction of fractions during the division of terms. In the exercise:

• The result \( \frac{1}{3}x \) arises from dividing \( 5 \) by \( 15 \). The fraction \( \frac{5}{15} \) reduces to \( \frac{1}{3} \).
• Similarly, \( \frac{-2}{3} \) is derived from \( \frac{-10}{15} \).
• Lastly, \( \frac{1}{3x} \) follows the same fractional reduction logic with \( \frac{5}{15} \).

By reducing fractions, we maintain the integrity of the expression while making each term simpler and clearer. Managing fractional coefficients is crucial for easing the simplification of algebraic expressions.