Problem 72 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{4 x^{2}+24 x+32}{16 x^{2}+8 x-48} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x + 4}{2(2x - 3)}\).

Step 1: Identify and Factor the Numerator

The numerator is \(4x^2 + 24x + 32\). We look for a common factor in all terms. Notice that each term is divisible by 4. Factoring 4 out, we get: \(4(x^2 + 6x + 8)\). Next, we need to factor the quadratic \(x^2 + 6x + 8\) further. Find two numbers that multiply to 8 and add to 6. These numbers are 2 and 4. Thus, the factorization is \(4(x + 2)(x + 4)\).

Step 2: Identify and Factor the Denominator

The denominator is \(16x^2 + 8x - 48\). First, factor out the greatest common factor, which is 8: \(8(2x^2 + x - 6)\). Now, we need to factor the quadratic \(2x^2 + x - 6\). Look for two numbers that multiply to \(2 \times (-6) = -12\) and add to 1. These numbers are 4 and -3. Replace the middle term with these numbers: \(2x^2 + 4x - 3x - 6\). Factor by grouping: \((2x^2 + 4x) - (3x + 6)\) gives \(2x(x + 2) - 3(x + 2)\). Factor out the common factor \((x + 2)\), yielding \((2x - 3)(x + 2)\).

Step 3: Write the Factored Form of Expression

Plug back the factored numerator and denominator into the original expression: \(\frac{4(x + 2)(x + 4)}{8(x + 2)(2x - 3)}\).

Step 4: Simplify the Expression

Cancel out the common factor \((x + 2)\) in the numerator and the denominator. This leaves us with: \(\frac{4(x + 4)}{8(2x - 3)}\). Simplify further by factoring out and canceling the common factor 4: \(\frac{(x + 4)}{2(2x - 3)}\).

Step 5: Final Step: Final Simplified Expression

The simplified expression is \(\frac{x + 4}{2(2x - 3)}\)

Key Concepts

Factoring Quadratics

Factoring quadratics is a technique used to simplify expressions or solve equations where the degree of the polynomial is two. In this process, you express a quadratic polynomial as a product of its linear factors. This method is particularly helpful when simplifying fractions like the one given in the original exercise. To factor a quadratic in the form of \(ax^2 + bx + c\), you need to find two numbers that multiply together to give \(ac\) (the product of the coefficient of \(x^2\) and the constant term), and sum to \(b\), the coefficient of \(x\). Let's see a quick recap of how this works using part of the exercise:

• Factor the quadratic numerator: \(x^2 + 6x + 8\)
• Find two numbers: 2 and 4, that multiply to 8 and add to 6
• Rewrite and factor: \((x + 2)(x + 4)\)

Understanding how to efficiently tackle these steps will enable you to simplify complex expressions or solve equations with ease.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers. In algebra, finding the GCF is a crucial step in simplifying expressions or factoring polynomial terms. It helps to break down expressions into simpler terms by finding common factors in numerical terms. In the context of polynomial expressions, the GCF is the highest term that can be factored out of each term in the polynomial. In the original exercise, the GCF was used to simplify both the numerator and the denominator by:

• Identifying common factors in the terms of the polynomial
• Dividing each term by the GCF
• Rewriting the polynomial with the GCF factored out

By incorporating the GCF into your simplifying strategy, you can reduce expressions to their simplest forms efficiently and effectively.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying rational expressions often involves factoring techniques, cancellation of common factors, and understanding the relationships between polynomials in the numerator and the denominator. The rational expression in the exercise was simplified by deploying the following steps:

• Factoring both the numerator and the denominator completely
• Identifying common polynomial factors in both parts
• Canceling out these common factors to simplify the fraction

This method reduces a rational expression to its simplest form, making further calculations easier and more accurate. By consistently applying these principles, you can handle complex algebraic fractions seamlessly.