Problem 43 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{3 n^{2}+16 n-12}{7 n^{2}+44 n+12}\)

Step-by-Step Solution

Answer

The simplified form is \(\frac{3n - 2}{7n + 2}\).

Step 1: Factor the Numerator

The first step in simplifying a rational expression is to factor both the numerator and the denominator, if possible. For the numerator \(3n^2 + 16n - 12\), we need to find two numbers that multiply to \(3 \times -12 = -36\) and add up to \(16\). These numbers are \(18\) and \(-2\). Thus, we can write the numerator as \((3n^2 + 18n - 2n - 12)\). Factor by grouping: \(= 3n(n + 6) - 2(n + 6)\). Therefore, the factored form is \((3n - 2)(n + 6)\).

Step 2: Factor the Denominator

Next, we factor the denominator \(7n^2 + 44n + 12\). We need numbers that multiply to \(7 \times 12 = 84\) and add up to \(44\). These numbers are \(42\) and \(2\). Write the expression as \((7n^2 + 42n + 2n + 12)\). Use grouping: \(7n(n + 6) + 2(n + 6)\). Thus, the denominator factors to \((7n + 2)(n + 6)\).

Step 3: Simplify the Rational Expression

The rational expression is now \(\frac{(3n - 2)(n + 6)}{(7n + 2)(n + 6)}\). Since \((n + 6)\) is a common factor in both the numerator and the denominator, we can cancel it out. This simplifies the expression to \(\frac{3n - 2}{7n + 2}\).

Step 4: Write the Simplified Form

After canceling the common factors, the simplified form of the given expression is \(\frac{3n - 2}{7n + 2}\). Ensure that any values that make the original denominator zero are noted as restrictions on the domain (\(n eq -\frac{2}{7}, -6\)).

Key Concepts

Factoring Polynomials

Polynomials are expressions comprising variables and coefficients, connected by operations like addition, subtraction, and multiplication. Factoring a polynomial involves rewriting it as a product of simpler polynomial factors. This simplification is crucial because it helps in reducing complex expressions and solving polynomial equations efficiently.

To factor a polynomial like the given numerator, \(3n^2 + 16n - 12\), we need to find numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient. Here we need numbers that multiply to \(3 \times -12 = -36\) and add to \(16\). These numbers are \(18\) and \(-2\).

By grouping, which is a method where terms are arranged in pairs, we can factor the expression:

• Group: \((3n^2 + 18n) - (2n + 12)\)
• Factor out common terms: \(3n(n + 6) - 2(n + 6)\)

The factored form of the numerator becomes \((3n - 2)(n + 6)\). This approach simplifies the process of handling complex polynomials, eventually making rational expressions easier to simplify.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Simplifying a rational expression involves factoring both the numerator and the denominator to identify and cancel common factors.

In our exercise, the goal is to simplify the rational expression \(\frac{3n^2 + 16n - 12}{7n^2 + 44n + 12}\). After factoring both the numerator and the denominator, the expression becomes \(\frac{(3n - 2)(n + 6)}{(7n + 2)(n + 6)}\).

The common factor, \((n + 6)\), appears in both parts. By canceling this common factor, the expression simplifies to \(\frac{3n - 2}{7n + 2}\).

This simplification utilizes the properties of fractions, which allow us to reduce the expression while maintaining essential characteristics. Understanding how to work with rational expressions is key to solving algebraic problems efficiently.

Domain Restrictions

While simplifying rational expressions, it is crucial to identify domain restrictions. These restrictions are values for the variable that make the denominator zero, since division by zero is undefined.

In the original expression \(\frac{3n^2 + 16n - 12}{7n^2 + 44n + 12}\), both the numerator and the denominator are polynomials. To find the domain restrictions, focus on the denominator. The factored form \((7n + 2)(n + 6)\) implies that the denominator becomes zero when either \(7n + 2 = 0\) or \(n + 6 = 0\).

• Solving \(7n + 2 = 0\) gives \(n = -\frac{2}{7}\).
• Solving \(n + 6 = 0\) gives \(n = -6\).

Thus, the domain restriction for this expression is \(n eq -\frac{2}{7}\) and \(n eq -6\). Recognizing these restrictions is essential to accurately simplifying and using rational expressions while avoiding mathematical errors.