Problem 33 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{2 n^{2}+5 n-3}{n^{2}-9}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2n - 1}{n - 3} \).

Step 1: Identify the Numerator and Denominator

First, identify the numerator and denominator of the algebraic fraction given. The numerator is \( 2n^2 + 5n - 3 \) and the denominator is \( n^2 - 9 \).

Step 2: Factor the Denominator

Look for a pattern or identity that fits the denominator \( n^2 - 9 \). Notice that this matches the difference of squares formula: \((a^2 - b^2 = (a + b)(a - b))\). Apply the formula: \( n^2 - 9 = (n + 3)(n - 3) \).

Step 3: Factor the Numerator

We need to factor the quadratic trinomial \( 2n^2 + 5n - 3 \). Use either factorization by grouping or trial and error to find that it factors to \((2n - 1)(n + 3)\).

Step 4: Simplify by Canceling Common Factors

Now that you have factored both the numerator and the denominator: \( \frac{(2n - 1)(n + 3)}{(n + 3)(n - 3)} \). Cancel out the common factor \( n + 3 \) appearing in both the numerator and the denominator.

Step 5: Write the Simplified Expression

After canceling \( n + 3 \), you are left with \( \frac{2n - 1}{n - 3} \). This is the simplified form of the original fraction.

Key Concepts

Factoring Quadratics

Factoring quadratics is a powerful tool when dealing with polynomial expressions. Quadratics are generally in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Factoring is the process of finding two binomials that will multiply together to give the original quadratic equation. This technique simplifies complex expressions for easier handling.
For the expression \(2n^2 + 5n - 3\), look to factor it by trial and error or grouping. Begin by identifying two numbers that multiply to \(a \times c\) (i.e., \(2 \times -3 = -6\)) and add to \(b\) (i.e., 5). The numbers (+6) and (-1) fit the criteria. Split the middle term accordingly and perform grouping:

• \(2n^2 + 6n - n - 3\)
• Group terms: \((2n^2 + 6n) + (-n - 3)\)
• Factor each: \(2n(n + 3) - 1(n + 3)\)

You then factor out common terms, resulting in \((2n - 1)(n + 3)\), the factored form.

Difference of Squares

The Difference of Squares is a specific pattern used to factor quadratic expressions. It works on any binomial in the form \(a^2 - b^2\), which factors to \((a + b)(a - b)\). This formula is incredibly useful for quickly breaking down expressions that match its pattern without lengthy steps.
Take the expression \(n^2 - 9\); recognize it fits the pattern with \(a = n\) and \(b = 3\). Applying the Difference of Squares formula gives us:

• \((n + 3)(n - 3)\)

This transformation simplifies the denominator, making it easier to cancel similar terms when simplifying algebraic fractions.

Simplification of Fractions

The simplification of fractions involves reducing the expression to its most basic form. After factoring both the numerator and the denominator, common terms can be cancelled.
Using the considered algebraic fraction \(\frac{2n^2 + 5n - 3}{n^2 - 9}\):
1. The numerator factors to \((2n - 1)(n + 3)\). 2. The denominator factors to \((n + 3)(n - 3)\).
Notice the \((n + 3)\) in both the numerator and the denominator. These are common factors and can be cancelled out:

• \(\frac{(2n - 1)(n + 3)}{(n + 3)(n - 3)} = \frac{2n - 1}{n - 3}\)

This procedure leaves the fraction in its simplest form, \(\frac{2n - 1}{n - 3}\), which is easier to work with in further calculations.