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

Step-by-Step Solution

Answer

The simplified expression is \(x^5 + \frac{3}{2}x^2\).

Step 1: Simplify the Numerator

The expression is \(\frac{2x^6 + 3x^3}{2x}\). Before applying the property, examine the numerator. It is already in expanded form as \(2x^6 + 3x^3\).

Step 2: Apply the Property

Use the property \(\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}\) to split the expression into two separate fractions: \(\frac{2x^6}{2x} + \frac{3x^3}{2x}\).

Step 3: Simplify Each Fraction

Simplify \(\frac{2x^6}{2x}\) by dividing the coefficients and subtracting the exponents: \(\frac{2}{2}x^{6-1} = x^5\). Similarly, simplify \(\frac{3x^3}{2x}\) by dividing the coefficients and subtracting the exponents: \(\frac{3}{2}x^{3-1} = \frac{3}{2}x^2\).

Step 4: Combine the Simplified Fractions

Combine the simplified expressions from the previous step, \(x^5 + \frac{3}{2}x^2\), to form the fully simplified expression.

Key Concepts

Fraction Decomposition

Fraction decomposition is the process of breaking down a complex fraction into simpler pieces. In algebra, this means rewriting a single fraction that has multiple terms in the numerator into several smaller fractions. This technique can often make calculations easier and reveal hidden simplifications.

In many algebraic expressions, like the one you've encountered, this method starts by observing the numerator. Consider the expression \(\frac{2x^6 + 3x^3}{2x}\). Notice that the numerator, \(2x^6 + 3x^3\), can be split into two fractions:

• \(\frac{2x^6}{2x}\)
• \(\frac{3x^3}{2x}\)

By decomposing the fraction, each part can be tackled individually, which allows for easier handling using other simplification techniques. This step is crucial because it sets up the next stages of simplification, where each smaller expression can be processed on its own. Understanding fraction decomposition fully helps in breaking down more complex algebraic problems into manageable units.

Polynomial Division

Polynomial division is an essential skill when simplifying algebraic expressions. It involves dividing terms within a polynomial, often using like bases to simplify further.

In the expression \(\frac{2x^6}{2x}\), we approach division by separating the coefficients and the variables. The coefficient \(2\) in the numerator is divided by the \(2\) in the denominator, giving us \(1\). This simplifies our focus to the variable terms, allowing us to subtract the exponents:

• \(x^{6-1} = x^5\)

Likewise, for \(\frac{3x^3}{2x}\), we divide \(3\) by \(2\), which gives \(\frac{3}{2}\), and for the variable, \(x^{3-1} = x^2\). When dividing polynomials, always remember to:

• Divide the numerical coefficients.
• Subtract the exponents of like bases.

Understanding these steps in polynomial division is key to simplifying expressions effectively and avoiding common mistakes.

Exponent Rules

Exponent rules are fundamental in simplifying expressions involving variables with powers. These rules help manipulate the expressions by operating on the powers while maintaining mathematical correctness.

In our example, these rules come into play when we simplified each fraction:

• For the term \(x^6\), divided by \(x^1\), we subtract the exponents (i.e., \(6 - 1\)), resulting in \(x^5\).
• Similarly, for \(x^3\) divided by \(x^1\), we perform the subtraction \(3 - 1\), yielding \(x^2\).

This use of exponent rules helps keep calculations straightforward by turning complex operations into simple subtractions. Here are the primary exponent rules to remember:

• Product of powers rule: \(x^a \cdot x^b = x^{a+b}\).
• Quotient of powers rule: \(\frac{x^a}{x^b} = x^{a-b}\).
• Power of a power rule: \((x^a)^b = x^{a \cdot b}\).

By mastering these rules, you can efficiently simplify expressions and solve algebraic problems that involve variables with exponents.