Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the rational expressions. \(\frac{12 n^{2}-29 n-8}{28 n^{2}-5 n-3}\)

Step-by-Step Solution

Answer

\(\frac{3n + 1}{7n + 3}\) is the simplified expression.

Step 1: Factor the Numerator

Examine the numerator, which is given as a quadratic expression: \(12n^2 - 29n - 8\). Use factoring techniques to rewrite it as the product of two binomials. The factorization is: \( (3n + 1)(4n - 8) \) after checking using the methods of finding two numbers that multiply to \(12 \times -8 = -96\) and add to \(-29\).

Step 2: Factor the Denominator

Consider the denominator, \(28n^2 - 5n - 3\), and factor it into the product of two binomials. The factorization is \((7n + 3)(4n - 1)\) by finding numbers that multiply to \(28 \times -3 = -84\) and add to \(-5\).

Step 3: Simplify the Expression by Cancelling Common Factors

Identify any common factors between the numerator and the denominator. In this case, \((4n - 1)\) is a common factor. Cancel these terms from the numerator and denominator to simplify the expression: \(\frac{(3n + 1)(4n - 2)}{(7n + 3)(4n - 1)}\) becomes \(\frac{(3n + 1)}{(7n + 3)}\).

Step 4: Verify the Simplification

Review the steps to ensure no further simplification can be made. Since \( (3n + 1)\) and \((7n + 3)\) share no more common factors, the expression is in its simplest form.

Key Concepts

Factoring Techniques

Factoring techniques are powerful tools in algebra that help us break down expressions into simpler factors. When tackling complex rational expressions, understanding how to factor is crucial.

• To factor a quadratic expression, focus on finding two numbers that multiply to give the product of the leading coefficient and the constant term, while adding up to the middle coefficient.
• This method allows us to rewrite the quadratic as a product of two binomials.

Let's take the numerator of our example, which is the quadratic expression \(12n^2 - 29n - 8\). By applying factoring techniques, we reformed it into \((3n + 1)(4n - 8)\). This involved identifying numbers that multiply to \(-96\) and sum to \(-29\). With practice, factoring becomes an intuitive step in simplifying expressions.

Quadratic Expression

A quadratic expression is a polynomial where the highest degree of the variable is two. It generally takes the form \(ax^2 + bx + c\). In these expressions, the "ax^2" term is crucial as it determines the shape of the parabola when graphed.

• The first step in simplifying or solving problems involving quadratics is often to factor them, as it can reveal solutions or simplify a rational expression.
• Quadratic expressions can usually be factored into two binomial expressions.

Consider the expression from our example: \(28n^2 - 5n - 3\). This expression was factored into \((7n + 3)(4n - 1)\). Recognizing quadratic forms quickly aids in simplification and solving.

Simplification

Simplification is the process of reducing an expression to its most basic form. This involves canceling out terms that appear in both the numerator and the denominator of a rational expression.

• Start by examining the factored form of both the numerator and denominator.
• Identify and cancel any common binomial factors to simplify the expression.

In our example, the factored form \((3n + 1)(4n - 2)\) over \((7n + 3)(4n - 1)\) allowed us to cancel the common term \((4n - 1)\) effortlessly. This removal leaves us with the simplified expression \(\frac{3n + 1}{7n + 3}\), which is the simplest form of the original problem.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. These are common in algebraic equations and are often used to test our ability to manipulate expressions effectively.

• The simplification of algebraic fractions involves finding and canceling common factors in the numerator and denominator.
• Always verify that no further common factors exist after simplification.

In the exercise, we worked with the algebraic fraction \(\frac{12 n^{2}-29 n-8}{28 n^{2}-5 n-3}\). By factoring each component and canceling common terms, the expression was simplified to \(\frac{3n + 1}{7n + 3}\). This process is central to understanding algebraic fractions and their simplification.