Problem 47 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{4 x^{2}-10 x-6}{2 x^{2}+11 x+5}$$

Step-by-Step Solution

Answer

\(\frac{4x^2 - 10x - 6}{2x^2 + 11x + 5} = \frac{2x-3}{x+5}\).

Step 1 - Factor the Numerator

Factor the quadratic expression in the numerator \(-10x\) to break up and simplify: \(4x^2 - 10x - 6 = (2x-6)(2x+1)\).

Step 2 - Factor the Denominator

Factor the quadratic expression in the denominator \(11x\) to identify common roots: \(2x^2 + 11x + 5 = (2x+1)(x+5)\).

Step 3 - Simplify the Expression

Cancel out the common factors in the numerator and denominator: \[(2x-3)(2x+1)\] / \[(2x+1)(x+5)\] \(= \frac{2x-3}{x+5}\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a key skill in reducing rational expressions. A quadratic expression is a polynomial of the form \(ax^2 + bx + c\). To factor it, look for two numbers that multiply to give you \(a \cdot c\) and add to give you \(b\). For example, to factor \(4x^2 - 10x - 6\), you need to find two numbers that multiply to \(-24\) and add to \(-10\). These numbers are \(-12\) and \(2\). You then break up \(bx\) using these numbers and factor by grouping. The factored form of \(4x^2 - 10x - 6\) is \(2x - 3\) and \(2x + 1\).

Simplifying Expressions

Simplifying expressions involves rewriting them in a simpler form. After factoring the numerator and denominator of a rational expression, check for common factors. If both the numerator and denominator share a common factor, you can cancel this factor out. For example, in the given expression, the factored numerator is \((2x-3)(2x+1)\) and the factored denominator is \((2x+1)(x+5)\). Here, \(2x+1\) is a common factor. Canceling it out simplifies the expression to \(\frac{2x-3}{x+5}\). Make sure to double-check that there's nothing left to cancel out and that the simplified form is fully reduced.

Algebraic Fractions

Algebraic fractions are fractions where the numerator or the denominator (or both) are algebraic expressions (i.e., polynomials). Working with these requires understanding operations such as addition, subtraction, multiplication, and division, along with factoring. When reducing algebraic fractions:

• First, factor the numerator and the denominator.
• Then, look for and cancel any common factors.
• The goal is to get the simplest form of the fraction.

Remember to always check your final answer to ensure you've fully simplified the expression. The process not only helps with simplification but also enhances your algebra skills.