Problem 2 - 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. $$\frac{6 x^{4}}{2 x^{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \( 3x \).

Step 1: Simplify the Denominator

The expression is \( \frac{6x^4}{2x^3} \). First, we can simplify the fraction by dividing both the numerator and the denominator by the coefficient 2 in the denominator.

Step 2: Divide the Coefficients

Since the coefficients in the original fraction are 6 in the numerator and 2 in the denominator, divide 6 by 2 to get 3. This simplifies the coefficient part of the fraction to \( \frac{6}{2} = 3 \).

Step 3: Simplify the Variables

Subtract the exponent of \( x \) in the denominator from the exponent of \( x \) in the numerator: \[x^{4-3} = x^1 = x\]Therefore, the variable part of the fraction becomes \( x \).

Step 4: Combine the Results

Combine the simplified coefficient and variable results to write the entire expression: \[ 3x \]Thus, \( \frac{6x^4}{2x^3} = 3x \).

Key Concepts

Fraction Simplification

Fractions can often look complex, but simplifying them makes things clearer and helps us work with them more efficiently. When simplifying fractions, our goal is to reduce them to their simplest form, where the numerator and the denominator have no common factors.
To simplify a fraction, first, look for any common factors in both the numerator and the denominator. Use those to divide each term. For example, in the fraction \( \frac{6}{2} \), both 6 and 2 can be divided by 2, making the simplified form \( 3 \).

• Find the greatest common factor (GCF).
• Divide both terms by the GCF.
• Check your result for further simplification opportunities, if necessary.

By consistently applying this process, you can make algebraic expressions more manageable, and easier for further operations.

Exponent Rules

Exponent rules are essential when working with algebraic expressions involving powers, as they provide shortcuts to simplify expressions quickly and accurately. A core rule when dealing with division is that you subtract the exponents. This is called the Quotient Rule for Exponents.
For instance, when dividing \( x^4 \) by \( x^3 \), subtract the exponent in the denominator from the exponent in the numerator: \[x^{4-3} = x^1 = x\]
Here’s why exponent rules are useful:

• Simplify expressions by reducing the number of terms.
• Make complex calculations quicker and easier.
• Help in recognizing patterns and equivalent expressions.

By mastering these rules, you can streamline calculations and ensure accurate results when working with powers.

Polynomial Division

Polynomial division allows us to simplify algebraic expressions when one polynomial is divided by another. It’s similar to arithmetic division but involves variables with exponents. One of the simplest forms is dividing monomials.
For instance, take \( \frac{6x^4}{2x^3} \). This involves:

• Dividing the coefficients (\( \frac{6}{2} = 3 \)).
• Subtracting the exponents of the like terms (variables):
  \[ x^{4-3} = x^1 = x \]

These two steps provide a comprehensive way to simplify any monomial division, transforming something potentially cumbersome into a straightforward result. Understanding and applying polynomial division is crucial in algebra because it helps solve more complex equations and facilitates factoring, simplifying polynomials, and even calculus operations.