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

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5x + 1}{3} \).

Step 1: Separate Terms

Start by applying the property \( \frac{a+b}{c}=\frac{a}{c}+\frac{b}{c} \). In our expression, \( a = 5x^3 \) and \( b = x^2 \), and \( c = 3x^2 \). Thus, break it into separate fractions: \[ \frac{5x^3}{3x^2} + \frac{x^2}{3x^2}. \]

Step 2: Simplify First Fraction

Simplify the first fraction \( \frac{5x^3}{3x^2} \) by cancelling common factors. Notice that \( x^3 = x^2 \cdot x \), so you can cancel \( x^2 \): \[ \frac{5x^3}{3x^2} = \frac{5x^2 \cdot x}{3x^2} = \frac{5x}{3}. \]

Step 3: Simplify Second Fraction

Simplify the second fraction \( \frac{x^2}{3x^2} \). Since \( x^2 \) appears in both the numerator and denominator, cancel it out: \[ \frac{x^2}{3x^2} = \frac{1}{3}. \]

Step 4: Combine Simplified Terms

Now, combine the simplified fractions from steps 2 and 3: \[ \frac{5x}{3} + \frac{1}{3}. \] Combine these since they have the same denominator: \[ \frac{5x + 1}{3}. \]

Key Concepts

Fraction Simplification

Fraction simplification involves breaking down complex fractions into their simplest form. It's a key skill in algebra, allowing you to work with more manageable expressions. Here, the main idea is to reduce fractions by cancelling common factors in the numerator and the denominator. For a given fraction, you look at each part and determine whether there are common numbers or variables that can be divided out of both the top and bottom.

In our original exercise, we start with a fraction like \(\frac{5x^3 + x^2}{3x^2}\). The first thing to do is separate this into two parts using the property \(\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}\):

• \(\frac{5x^3}{3x^2} + \frac{x^2}{3x^2}\).

Each part is simplified individually:

• For \(\frac{5x^3}{3x^2}\), you divide both the numerator and denominator by \(x^2\), leaving you with \(\frac{5x}{3}\).
• For \(\frac{x^2}{3x^2}\), divide by \(x^2\), simplifying it to \(\frac{1}{3}\).

Fraction simplification is crucial to make expressions easier to handle, especially in more complex problems.

Polynomial Division

Polynomial division entails dividing expressions that include variables raised to different powers. The goal is to simplify the problem by breaking it down in a systematic way, identifying parts of the polynomials that can be reduced or cancelled out.

In this exercise, the given polynomial in the numerator was split into two fractions. Therefore, you're able to simplify both terms individually. This demonstrates the method of polynomial division without actual long division.

When dividing polynomials, keep an eye on exponents:

• Identify common variables or factors in the numerator and denominator.
• Cancel these common factors where possible.

In the example, we had \(5x^3\) and \(x^2\) in the numerator and \(3x^2\) in the denominator. By splitting them and dividing, we achieved simplified terms like \(\frac{5x}{3}\) and \(\frac{1}{3}\). It’s a simple yet powerful method!

Algebraic Techniques

Algebraic techniques are methods and strategies used to simplify and solve algebraic expressions and equations. These techniques include properties of operations, factorization, and expression manipulation.

The method applied in the given exercise involved logical breakdown of expressions, strategic cancellation of terms, and combining results. In particular:

• Using properties like \(\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}\) to separate terms simplifies complex expressions to more workable parts.
• Cancelling out common terms or factors reduces expressions to their simplest form, making further operations or problem-solving clearer.
• Combining results from several steps results in cleaner, more manageable expressions.

By applying these algebraic techniques, you not only solve the problem at hand but also gain a better understanding of how expressions are structured and how to tackle similar problems in the future.