Problem 48 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{6 x^{2}+x-1}{8 x^{2}-2 x-3}$$

Step-by-Step Solution

Answer

\(\frac{3x - 1}{4x - 3}\)

Step 1 - Factorize the numerator

First, factorize the numerator of the rational expression. The numerator is \(6x^2 + x - 1\). We need to find two numbers that multiply to \(6 \times -1 = -6\) and add to \(1\). These numbers are \(3\) and \(-2\). So, we can rewrite the middle term and factor by grouping: \(6x^2 + 3x - 2x - 1\). Group the terms: \((6x^2 + 3x) - (2x + 1)\). Factor each group: \(3x(2x + 1) - 1(2x + 1)\). Combine common factors: \((3x - 1)(2x + 1)\).

Step 2 - Factorize the denominator

Next, factorize the denominator of the rational expression. The denominator is \(8x^2 - 2x - 3\). We need to find two numbers that multiply to \(8 \times -3 = -24\) and add to \(-2\). These numbers are \(4\) and \(-6\). So, we can rewrite the middle term and factor by grouping: \(8x^2 - 6x + 4x - 3\). Group the terms: \((8x^2 - 6x) + (4x - 3)\). Factor each group: \(2x(4x - 3) + 1(4x - 3)\). Combine common factors: \((2x + 1)(4x - 3)\).

Step 3 - Simplify the fraction

Combine the factored forms of the numerator and the denominator: \(\frac{(3x-1)(2x+1)}{(2x+1)(4x-3)}\). Cancel out the common factors in the numerator and the denominator: \(\frac{3x - 1}{4x - 3}\).

Key Concepts

Factoring Polynomials

Understanding how to factorize polynomials is essential in algebra. Polynomials are expressions that include terms with variables raised to whole number exponents. To factor a polynomial is to break it down into simpler expressions (its factors) that, when multiplied, give back the original polynomial.

Here's a quick guide to factorizing polynomials:

• Identify common factors in all terms and factor them out.
• Use techniques like grouping to manage more complicated polynomials.
• Search for pairs of terms that, when multiplied, give the original polynomial and then group and factor them separately.

For example, in the polynomial \(6x^2 + x - 1\), we identify and group terms to factorize: \begin{align*} 6x^2 + x - 1 &= 6x^2 + 3x - 2x - 1 \ &= 3x(2x + 1) - 1(2x + 1) \ &= (3x - 1)(2x + 1)\tag*{} \tag*{} \tag*{}\tag*{}\tag*{}\tag*{}\tag*{}\tag*{}\tag*{} \tag*{}\tag*{}\tag*{}\tag*{}\tag*{} By spotting and grouping terms correctly, you can efficiently factor many polynomials.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Working with rational expressions involves several steps, especially factoring and simplifying.

First, always look to factorize the numerator and the denominator. Once you have the factors, it becomes easier to simplify. Consider the rational expression \(\frac{6x^2 + x - 1}{8x^2 - 2x - 3}\).
We factorize:

• Numerator: \(6x^2 + x - 1 = (3x - 1)(2x + 1)\)
• Denominator: \(8x^2 - 2x - 3 = (2x + 1)(4x - 3)\)

Now, the expression looks like \(\frac{(3x - 1)(2x + 1)}{(2x + 1)(4x - 3)}\). By canceling out the common factors \((2x + 1)\), we simplify it to \(\frac{3x - 1}{4x - 3}\).

Simplifying Fractions

Simplifying fractions, whether numeric or algebraic, is a fundamental skill. It makes complex problems easier to solve and understand. To simplify a fraction, you divide both the numerator and the denominator by their greatest common factor (GCF).

In algebra, simplifying involves:

• Factoring both the numerator and the denominator.
• Canceling common factors between them.

For example, given the rational expression \(\frac{6x^2 + x - 1}{8x^2 - 2x - 3}\), we factor both parts: \begin{align*} \text{Numerator:} & (3x - 1)(2x + 1) \ \text{Denominator:} & (2x + 1)(4x - 3) \tag*{} \tag*{}\tag*{}\tag*{}<> Equating them, and canceling the common factor \((2x + 1)\), we simplify to \(\frac{3x - 1}{4x - 3}\). Always check for any factor that can divide both the numerator and the denominator to their simplest forms.

Algebraic Techniques

Mastering algebra involves a variety of techniques, especially when working with rational expressions. These include:

• Factoring: Breaking down expressions into products of simpler ones.
• Grouping: A method where you group terms to simplify the factoring process.
• Canceling Common Factors: Simplifying by removing factors that appear in both the numerator and the denominator.

Combining these techniques helps solve complicated algebraic problems more manageably. For instance, in \(\frac{6x^2 + x - 1}{8x^2 - 2x - 3}\), by:

• Factoring both the numerator and the denominator,
• Recognizing and canceling out common terms,
• Reducing the expression to its simplest form \(\frac{3x - 1}{4x - 3}\).

Building proficiency in these techniques is crucial for tackling higher-level math problems efficiently and accurately.