Problem 9 - College Algebra

Question

Simplify the rational expressions. $$ \frac{2 x^{2}+7 x-4}{4 x^{2}+2 x-2} $$

Step-by-Step Solution

Answer

\(\frac{x + 4}{2(x + 1)}\)

Step 1: Factor the Numerator

We start by factoring the quadratic expression in the numerator: \(2x^2 + 7x - 4\). We are looking for two numbers that multiply to \(2 \times -4 = -8\) and add to \(7\). These numbers are \(8\) and \(-1\). The factorization is:\[2x^2 + 7x - 4 = (2x - 1)(x + 4)\]

Step 2: Factor the Denominator

Now, let's factor the quadratic expression in the denominator: \(4x^2 + 2x - 2\). We are looking for two numbers that multiply to \(4 \times -2 = -8\) and add to \(2\). These numbers are \(4\) and \(-2\). The factorization is:\[4x^2 + 2x - 2 = (2x - 1)(2x + 2)\]

Step 3: Simplify the Expression

Now that we have factored both the numerator and the denominator, we check for common factors. We notice that \(2x - 1\) appears in both the numerator and the denominator. Thus, we can simplify the expression by canceling the \(2x - 1\) term:\[\frac{(2x - 1)(x + 4)}{(2x - 1)(2x + 2)} = \frac{x + 4}{2x + 2}\]

Step 4: Further Simplify

Let's simplify the denominator. We can factor \(2x + 2\) as \(2(x + 1)\). Substituting this in gives the expression:\[\frac{x + 4}{2(x + 1)}\]

Step 5: Conclusion

The rational expression \(\frac{2x^2 + 7x - 4}{4x^2 + 2x - 2}\) simplifies to \(\frac{x + 4}{2(x + 1)}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential part of working with rational expressions. It involves reducing the fraction to its simplest form by eliminating common factors from the numerator and the denominator. To do this:

• Identify common factors shared by the numerator and the denominator.
• Cancel out these common factors to simplify the expression.
• Ensure that the final form of the fraction cannot be reduced further.

In the given problem, after factoring the expressions, we found that both the numerator and the denominator contained the common factor \(2x - 1\). By canceling this out, the fraction is simplified to \(\frac{x + 4}{2(x + 1)}\). This ensures we work with the simplest version of the rational expression, making it easier to understand and work with in further calculations.

Factoring Quadratics

Factoring quadratics is a method used to break down quadratic expressions into simpler, multiplied components. This technique can reveal common factors, which are necessary for simplifying rational expressions.

• Start by writing the quadratic in the form \(ax^2 + bx + c\).
• Look for two numbers that multiply to \(ac\) (product of the coefficient of \(x^2\) and the constant \(c\)) and add up to \(b\) (the coefficient of \(x\)).
• Rearrange the quadratic using these two numbers and factor by grouping.

In the problem, we first factored the numerator, finding \((2x - 1)(x + 4)\). Similarly, the denominator was factored into \((2x - 1)(2x + 2)\). Mastery of this technique allows for effective simplification of more complex rational expressions.

Numerator and Denominator

The numerator and denominator are the top and bottom parts of a fraction, respectively. They are crucial in the formation of rational expressions—where the numerator is divided by the denominator. Understanding their roles is essential for tasks like simplifying and factoring.

• The numerator is the expression above the division line in a fraction.
• The denominator is the expression below the division line, indicating the part into which the numerator is divided.
• For simplification, finding common elements in both is key.

In our example, the numerator \(2x^2 + 7x - 4\) and denominator \(4x^2 + 2x - 2\) were individually factored to reveal common factors. This insight allows us to effectively simplify the rational expression.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks for equations and functions in algebra. Understanding how to manipulate and simplify these expressions is crucial in algebra.

• They can include variables raised to powers, constants, and various operations such as addition and subtraction.
• Common tasks include simplifying, factoring, expanding, and solving equations involving algebraic expressions.
• Recognizing patterns and factors within these expressions helps in tackling more advanced problems.

In the problem at hand, the expressions \(2x^2 + 7x - 4\) and \(4x^2 + 2x - 2\) exemplify algebraic expressions. The goal was to manipulate these expressions through factoring and simplification, illustrating deeper insights into algebra and rational expressions.