Problem 48 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{6 n-60}{n^{2}-20 n+100}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{6}{n-10} \).

Step 1: Factor the Numerator

The numerator of the fraction is \( 6n - 60 \). Notice that both terms in the numerator can be factored by 6. This results in \( 6(n - 10) \).

Step 2: Factor the Denominator

The denominator is \( n^2 - 20n + 100 \). This is a perfect square trinomial because \( n^2 - 20n + 100 = (n - 10)^2 \).

Step 3: Simplify the Fraction

Replace the numerator and the denominator of the fraction with their factored forms: \( \frac{6(n-10)}{(n-10)(n-10)} \). Since \( n - 10 \) is a common factor in both the numerator and the denominator, it can be canceled from the fraction.

Step 4: Final Simplification

After canceling the \( n - 10 \) term in both the numerator and the denominator, we are left with \( \frac{6}{n-10} \). This is the simplified form of the original algebraic fraction.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial expression as a product of its factors. This is an essential skill in simplifying algebraic fractions, solving polynomial equations, and more.
To effectively factor a polynomial:

• Look for any common factors in all the terms of the polynomial.
• Check if the polynomial can be expressed as a product of simpler polynomials.
• Use techniques such as grouping, the difference of squares, or recognizing special forms like perfect square trinomials and cubes.

In the exercise provided, we first factor the numerator by identifying the factor common to both terms, 6. This reduces the polynomial expression into a simpler form: \(6(n - 10)\).
This is just one example of factoring and can be accomplished for many types of polynomial expressions with practice.

Perfect Square Trinomial

A perfect square trinomial is a specific type of polynomial that can be written as the square of a binomial. These trinomials have distinctive features that allow us to recognize and factor them easily.
The general form of a perfect square trinomial is \(a^2 \pm 2ab + b^2\), which factors into \((a \pm b)^2\).
In the context of our exercise, the denominator \(n^2 - 20n + 100\) follows this pattern:

• Identify the square roots of the first and last terms: \(n\) and 10.
• Check that the middle term matches \(-2 \times n \times 10\), which it does at \(-20n\).

Thus, \(n^2 - 20n + 100\) is indeed a perfect square trinomial, simplifying to \((n - 10)^2\). Recognizing such patterns helps streamline the factoring process, making it quicker and more efficient.

Common Factors

Common factors play a crucial role in simplifying fractions because they can be divided out, significantly reducing the complexity of expressions. When both the numerator and denominator of a fraction have a common factor, this factor can be canceled out from the fraction.
To identify and use common factors:

• Factor the numerator and the denominator completely.
• Look for factors that appear in both the factored forms.
• Cancel these common factors to simplify the fraction.

In our problem, the factor \(n-10\) is common to both the numerator \(6(n-10)\) and the denominator \((n-10)(n-10)\). Canceling \(n-10\) from the top and bottom leads to the simpler expression \(\frac{6}{n-10}\).
By identifying and eliminating common factors, the process of simplification becomes straightforward and highly effective.