Problem 15 - A Graphical Approach to College Algebra

Question

Write each rational expression in lowest terms. $$\frac{8 x^{2}+16 x}{4 x^{2}}$$

Step-by-Step Solution

Answer

\(\frac{2(x+2)}{x}\)

Step 1: Factor the Numerator

First, we factor the numerator of the expression \(8x^2 + 16x\) by finding the greatest common factor (GCF). The GCF of \(8x^2\) and \(16x\) is \(8x\). Factoring out \(8x\), we have: \(8x(x + 2)\).

Step 2: Factor the Denominator

Next, we need to consider the denominator, \(4x^2\). The expression \(4x^2\) is already factored, consisting of \(4\) and \(x^2\).

Step 3: Cancel Common Factors

Now, we identify and cancel any common factors between the numerator and the denominator. Both \(8x(x + 2)\) and \(4x^2\) have a common factor of \(4x\). Dividing by \(4x\), we get: \[\frac{8x(x + 2)}{4x^2} \rightarrow \frac{2(x + 2)}{x}\].

Step 4: Simplify the Expression

After cancelling the common factor, we're left with \(\frac{2(x + 2)}{x}\), which is the simplified form of the original rational expression.

Key Concepts

Factoring

Factoring is a key technique in simplifying rational expressions. By rewriting a polynomial as a product of simpler polynomials, it's possible to reveal common factors in expressions that can be simplified further. For example, in the numerator of the given expression, we have the polynomial \(8x^2 + 16x\).

To factor it, we look for terms common to both parts of the polynomial. This involves identifying the greatest number and power of \(x\) common to each term. Factoring makes it easier to reduce the rational expression by canceling out these common factors later on. Once you have factored out the greatest common factor, always double-check by redistributing to ensure you haven’t missed any common factors.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest term that divides each term of a polynomial without leaving a remainder. It's a vital step in simplifying expressions as it allows breaking down the expression into simpler parts.

Take the expression \(8x^2 + 16x\). Here, the GCF is found by considering both the numeric coefficients and the variable parts.

• The greatest number that divides 8 and 16 is 8.
• The smallest exponent of \(x\) in "\(x^2\)" and "\(x\)" is 1, so the GCF in terms of \(x\) is \(x\).

Thus, the GCF of \(8x^2\) and \(16x\) is \(8x\). Factoring out \(8x\) simplifies the polynomial and uncovers the structure of the expression.

Cancel Common Factors

Canceling common factors is a key step in simplifying a rational expression. Once both the numerator and the denominator are factored, we identify any factors common to both.

This is possible due to the property that any number divided by itself equals 1.

• Start by factoring both the numerator and the denominator completely.
• Identify common factors and cancel them. Remember to cancel entire factors, not individual terms within them.
• In the exercise at hand, both the numerator \(8x(x + 2)\) and the denominator \(4x^2\) share a common factor, \(4x\). By canceling \(4x\), you arrive at the simplified form, \(\frac{2(x + 2)}{x}\).

This step is crucial because it simplifies complex expressions, making them easier to understand and work with.

Algebraic Fractions

Rational expressions, also known as algebraic fractions, function similarly to traditional numerical fractions, except they involve polynomials. By simplifying these expressions, you can better solve equations and understand the relationships they represent.

To simplify an algebraic fraction:

• Factor both the numerator and denominator completely.
• Identify and cancel any common factors.
• Reduce the resulting expression to its simplest form.

Understanding algebraic fractions allows you to manipulate and solve equations involving variables effectively, akin to how understanding basic fractions facilitates numerical calculations. It’s about breaking down complex expressions into simpler, manageable parts and finding their simplest form.