Problem 11 - A Graphical Approach to College Algebra

Question

Write each rational expression in lowest terms. $$\frac{8 k+16}{9 k+18}$$

Step-by-Step Solution

Answer

The simplified form is \(\frac{8}{9}\).

Step 1: Identify the Greatest Common Factor (GCF) of the Numerator

The first step is to factor out the greatest common factor (GCF) from the numerator, which is \(8k + 16\). The GCF of \(8k\) and \(16\) is \(8\). So we factor \(8\) out: \(8k + 16 = 8(k + 2)\).

Step 2: Identify the Greatest Common Factor (GCF) of the Denominator

Next, factor out the GCF from the denominator, \(9k + 18\). The GCF of \(9k\) and \(18\) is \(9\). So we factor \(9\) out: \(9k + 18 = 9(k + 2)\).

Step 3: Simplify the Expression

Now that both the numerator and the denominator are factored, the expression is \(\frac{8(k + 2)}{9(k + 2)}\). We can simplify the expression by canceling out the common factor \((k + 2)\) from the numerator and the denominator, resulting in \(\frac{8}{9}\).

Step 4: Verify the Simplified Expression

Ensure that \(k + 2\) is not equal to zero and is not a factor left in the expression, as it was canceled during simplification. The expression is valid for all \(k\) not equal to \(-2\) due to the cancellation step.

Key Concepts

Greatest Common Factor (GCF)

In algebra, identifying the Greatest Common Factor (GCF) is crucial when simplifying expressions. The GCF is the largest factor that divides all terms in an expression without leaving a remainder.

Understanding the GCF helps to break down complex polynomials into simpler terms, making them easier to manage.

• How to find it: List all factors of each term in the expression. The highest number that appears in all lists is the GCF.


• Example: In the expression \(8k + 16\), the GCF is 8 because both 8 and 16 are divisible by 8.
  

Factoring out the GCF simplifies the arithmetic and helps in further steps.

For any number, understanding common factors assists in unlocking the potential to reduce and simplify algebraic expressions fundamentally.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This is especially useful when simplifying rational expressions.

To factor polynomials effectively, begin by finding the GCF, and factor it out from the expression.

• Steps: Identify the GCF. Once found, divide each term by the GCF, writing the polynomial as a product of the GCF and the resulting expression.


• Example: For \(8k + 16\), we identified the GCF as 8 in the previous section. When we factor 8 out, we have \(8(k + 2)\).

This step is essential as it sets the polynomial for further simplification.

Factoring not only helps in simplification but also plays a key role in solving polynomial equations, finding roots, and understanding their expressions.

Simplifying Algebraic Fractions

Simplifying algebraic fractions is a process where the fraction is reduced to its simplest form. This requires both the numerator and the denominator to be factored first.

Cancellation of common factors in the numerator and denominator is a key step. Here's how:

• First, identify and factor out the GCF from both the numerator and denominator.


• Next, write the expression as a fraction of these factors.


• Cancel out any common terms (provided it's not zero or does not render the expression undefined).


• Example: In the expression \(\frac{8(k+2)}{9(k+2)}\), \((k+2)\) is a common factor. Canceling \((k+2)\) leaves \(\frac{8}{9}\).

Through simplification, the fraction becomes much more manageable and easier to interpret.

Remember, the aim is to ensure the expression is valid for all possible values of the variable, except those that make any original denominator zero.