Problem 40 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{n^{2}-9}{n^{2}+4 n+3} $$

Step-by-Step Solution

Answer

\( \frac{n - 3}{n + 1} \)

Step 1: Factor the Numerator

Factor the numerator, which is a difference of squares. Recognize that the numerator can be written as: \[ n^2 - 9 = (n + 3)(n - 3) \]

Step 2: Factor the Denominator

Factor the denominator, which is a quadratic expression. Identify two numbers that multiply to 3 and add to 4. The factorization is: \[ n^2 + 4n + 3 = (n + 3)(n + 1) \]

Step 3: Write the Fraction with Factored Numerator and Denominator

Rewrite the original fraction using the factored forms from Steps 1 and 2: \[ \frac{n^2 - 9}{n^2 + 4n + 3} = \frac{(n + 3)(n - 3)}{(n + 3)(n + 1)} \]

Step 4: Simplify the Fraction

Cancel the common factor \(n + 3\) from the numerator and denominator:\[ \frac{(n + 3)(n - 3)}{(n + 3)(n + 1)} = \frac{n - 3}{n + 1} \]

Key Concepts

Factoring Quadratics

Factoring quadratics is a fundamental step in simplifying algebraic fractions. A quadratic expression is typically in the form of \(ax^2 + bx + c\). To factor it, you need to find two numbers that multiply to \(ac\) and add to \(b\). In our example, the quadratic expression in the denominator is \(n^2 + 4n + 3\). Here, \(a = 1\), \(b = 4\), and \(c = 3\). We look for two numbers that multiply to \(1 \times 3 = 3\) and add to \(4\). These numbers are \(3\) and \(1\). Therefore, \(n^2 + 4n + 3\) factors to \( (n + 3)(n + 1) \). When teaching factoring quadratics:\- Break down the steps gradually.\- Practice with different examples.\- Emphasize the importance of understanding each value's role.

Difference of Squares

Recognizing and factoring the difference of squares can simplify fractions significantly. The difference of squares formula is \[a^2 - b^2 = (a + b)(a - b)\]. In the given problem, the numerator \(n^2 - 9\) is a perfect candidate for this method. Here, \(n^2\) is \(a^2\) and \(9\) is \(b^2\) (where \(b = 3\)). Hence, \[n^2 - 9 = (n + 3)(n - 3)\], making it easier to work with and eventually simplifying the algebraic fraction. To master this formula:\- Understand the structure of squares (which numbers are squares of others).\- Practice with multiple examples.\- Remember that both terms must be perfect squares for this method to apply.

Canceling Common Factors

Canceling common factors is a crucial step in simplifying fractions. After factoring, you might find that the numerator and denominator share common factors. By removing these, you simplify the expression. For instance, our fraction \[ \frac{(n+3)(n-3)}{(n+3)(n+1)}\] has \(n+3\) in both the numerator and denominator. We cancel them out, reducing the fraction to \[ \frac{n-3}{n+1}\]. Key points for canceling common factors: \- Ensure the factors are identical. \- Only cancel factors, not terms within a polynomial. \- Check for other possible cancellations after the initial factor removal.

Simplifying Rational Expressions

Simplifying rational expressions involves combining all the previous steps into a cohesive process. Start by factoring both the numerator and the denominator. Then cancel any common factors. This leaves you with a simplified expression. In our exercise, after factoring \(n^2 - 9\) into \( (n + 3)(n - 3) \) and \(n^2 + 4n + 3\) into \( (n + 3)(n + 1) \), we cancel the common factor \(n + 3\) and simplify to \[ \frac{n - 3}{n + 1}\]. Tips for simplifying rational expressions: \- Always factor first.\- Be cautious of non-permissible values that make the denominator zero.\- Re-check your final expression for any further possible simplifications.