Problem 9 - College Algebra

Question

For the following exercises, simplify the rational expressions. $$ \frac{2 x^{2}+7 x-4}{4 x^{2}+2 x-2} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{x + 4}{2(x + 1)} \).

Step 1: Factor the Numerator

The numerator of the expression is \(2x^2 + 7x - 4\). We need to factor it fully. Start by finding two numbers that multiply to \(2 \times -4 = -8\) and add up to \(7\). These numbers are \(8\) and \(-1\).\[2x^2 + 7x - 4 = 2x^2 + 8x - x - 4\]Next, group the terms:\[(2x^2 + 8x) + (-x - 4)\]Factor by grouping:\[2x(x + 4) - 1(x + 4)\]This gives the factorization:\[(2x - 1)(x + 4)\]

Step 2: Factor the Denominator

The denominator of the expression is \(4x^2 + 2x - 2\). Factor this expression:Start by simplifying:\[2(2x^2 + x - 1)\]Now, find two numbers that multiply to \(2 \times -1 = -2\) and add up to \(1\). These numbers are \(2\) and \(-1\).\[2x^2 + x - 1 = 2x^2 + 2x - x - 1\]Group the terms:\[(2x^2 + 2x) + (-x - 1)\]Factor by grouping:\[2x(x + 1) - 1(x + 1)\]Factor out the common term:\[(2x - 1)(x + 1)\]Thus, the factorized expression is \(2(2x-1)(x+1)\).

Step 3: Simplify the Expression

Now that both the numerator \((2x - 1)(x + 4)\) and the denominator \(2(2x - 1)(x + 1)\) are factored, simplify by canceling the common terms:- The common term is \((2x - 1)\).Cancel \((2x - 1)\):\[\frac{(2x - 1)(x + 4)}{2(2x - 1)(x + 1)} = \frac{x + 4}{2(x + 1)}\]

Step 4: State the Simplified Expression

After canceling the common factor, the simplified expression is:\[\frac{x + 4}{2(x + 1)}\] This is the simplest form obtainable from the given rational expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra that involves breaking down a polynomial into its simpler components, called factors. This process can make complex expressions more manageable and easier to work with. In this exercise, we start by factoring both the numerator and the denominator of a rational expression.

To factor a polynomial, we need to identify numbers or expressions that multiply together to give the original expression. For the numerator \(2x^2 + 7x - 4\), we first need two numbers that multiply to the product of the leading coefficient and the constant term, which, in this case, is \(-8\). These numbers are \(8\) and \(-1\), as they add up to \(7\), the coefficient of the linear term. Next, we rearrange and group the terms:

• Group the terms: \((2x^2 + 8x) + (-x - 4)\)
• Factor by grouping: \(2x(x + 4) - 1(x + 4)\)

Thus, the factored form of the numerator is \((2x - 1)(x + 4)\).Similarly, for the denominator \(4x^2 + 2x - 2\), we follow analogous steps to factor it, eventually leading to the factored form: \(2(2x - 1)(x + 1)\). Factoring makes it easier to simplify rational expressions by identifying common terms that can be canceled.

Rational Expressions

A rational expression is simply a fraction in which the numerator and the denominator are both polynomials. Simplifying rational expressions involves reducing these fractions to their simplest form.In our exercise, we have the rational expression:\[\frac{2x^2 + 7x - 4}{4x^2 + 2x - 2}\]Factoring both the numerator and the denominator reveals common terms that can be canceled. This step is crucial for simplifying the expression.Rational expressions behave similarly to regular fractions when it comes to simplification: Cancel out the common factors from the numerator and the denominator. In this case, after factoring, we find the common term \((2x - 1)\). Cancelling this term out reduces our expression significantly.

• Factor the numerator and denominator.
• Identify and cancel common terms.

After these steps, the rational expression becomes less complex and easier to understand and work with. It's essential to be comfortable with these techniques, as they are widely used in various mathematical applications.

Simplifying Fractions

Simplifying fractions, whether numerical or algebraic, involves reducing them to their simplest form by canceling out common factors. This is a fundamental concept in algebra that enhances clarity and ease of calculation. For rational expressions like \(\frac{2x^2 + 7x - 4}{4x^2 + 2x - 2}\), simplifying the expression is about canceling common factors after factoring both the numerator and the denominator.Here's what happens during simplification: the expression is expressed in terms of its factors, and common terms are canceled. The key steps include:

• Factor the numerator (\( (2x - 1)(x + 4) \)).
• Factor the denominator (\( 2(2x - 1)(x + 1) \)).
• Cancel the common factor \((2x - 1)\).

After canceling, the simplified expression is:\[\frac{x + 4}{2(x + 1)}\]This reduced form is easier to interpret and work with in further mathematical operations. Simplifying rational expressions is a practical skill that underscores the importance of factoring as a preliminary step in many algebraic processes. It ensures that calculations are not only correct but also efficient and straightforward.