Problem 76 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{x+2}{12 x^{3}}+\frac{x+1}{4 x^{2}+8 x-12}-\frac{x+3}{16 x^{2}-32 x+16} $$

Step-by-Step Solution

Answer

Answer: $$\frac{6x^4 - 3x^3x^2 + 3x^3 + 4x^2 + 12x - 8}{2^4 \cdot 3 \cdot x^3 \cdot (x+3)(x-1)}$$

Step 1: Factor the denominators of rational expressions

First, we need to factor the denominators of each rational expression. This will help us find the least common denominator (LCD) for all the expressions. For the first expression, the denominator is already factored: \(12x^3 = 2^2 \cdot 3 \cdot x^3\) For the second expression, first factor the common factor and then factor the remaining quadratic expression: $$ 4x^2 + 8x - 12 = 4(x^2+2x-3) = 4(x+3)(x-1) $$ For the third expression, first factor the common factor: $$ 16x^2 - 32x + 16 = 16(x^2-2x+1) = 16(x-1)^2 $$

Step 2: Determine the Least Common Denominator (LCD)

Now we need to find the LCD for all the denominators. The LCD will be the product of the highest powers of each unique factor from the three denominators. $$ \text{LCD} = 2^4 \cdot 3 \cdot x^3 \cdot (x+3)(x-1) $$

Step 3: Write each rational expression with the LCD as the denominator

Next, we need to rewrite each rational expression with the LCD as the denominator. To do this, we'll multiply each expression by the missing factors from the LCD. $$ \frac{x+2}{12 x^{3}}\cdot \frac{2^2 \cdot (x+3)(x-1)}{2^2 \cdot (x+3)(x-1)} + \frac{x+1}{4 (x+3)(x-1)}\cdot \frac{2^3\cdot 3\cdot x^3}{2^3\cdot 3\cdot x^3} - \frac{x+3}{16(x-1)^2} \cdot \frac{3x^3(x+3)}{3x^3(x+3)} $$

Step 4: Simplify and add/subtract the rational expressions

After rewriting each expression with the same denominator, we can combine the numerators and simplify the expression. $$ \frac{(x+2)(2^2)(x+3)(x-1) + (x+1)(2^3 \cdot 3 \cdot x^3) - (x+3)(3x^3)(x+3)}{2^4 \cdot 3 \cdot x^3 \cdot (x+3)(x-1)} $$ Now distribute and combine like terms: $$ \frac{4(x^2+3x-2) + 24x^3(x+1) - 3x^3(x+3)^2}{2^4 \cdot 3 \cdot x^3 \cdot (x+3)(x-1)} $$ After expanding and simplifying: $$ \frac{4x^2 + 12x - 8 + 24x^4 + 24x^3 - 3x^3(x^2+6x+9)}{2^4 \cdot 3 \cdot x^3 \cdot (x+3)(x-1)} $$ $$ \frac{4x^2 + 12x - 8 + 24x^4 + 24x^3 - 3x^3x^2 - 18x^4 - 27x^3}{2^4 \cdot 3 \cdot x^3 \cdot (x+3)(x-1)} $$ Combine like terms in the numerator: $$ \frac{24x^4 - 18x^4 + 4x^2 - 3x^3x^2 - 24x^3 + 27x^3 + 12x - 8}{2^4 \cdot 3 \cdot x^3 \cdot (x+3)(x-1)} $$ $$ \frac{6x^4 - 3x^3x^2 + 3x^3 + 4x^2 + 12x - 8}{2^4 \cdot 3 \cdot x^3 \cdot (x+3)(x-1)} $$ The final simplified expression is: $$ \frac{6x^4 - 3x^3x^2 + 3x^3 + 4x^2 + 12x - 8}{2^4 \cdot 3 \cdot x^3 \cdot (x+3)(x-1)} $$

Key Concepts

Factoring Polynomials

Factoring polynomials is a key mathematical technique used to simplify complex expressions. In many cases, this process involves finding the simplest set of expressions that, when multiplied together, equal the original polynomial. This is vital in operations involving rational expressions which often necessitate simplification or finding a common denominator.

For example, consider the polynomial:

• \(16x^2 - 32x + 16\)
• The first step is to factor a common factor. Here, you can factor out 16, resulting in \(16(x^2 - 2x + 1)\).
• This expression is a perfect square trinomial and can be further factored to \(16(x-1)^2\).

The ability to recognize patterns in polynomials such as the difference of squares or quadratics helps in effective factoring. This same rational approach is applied to all denominators in rational expressions, facilitating further steps like identifying the Least Common Denominator (LCD).

Least Common Denominator

Finding the Least Common Denominator (LCD) is crucial when working with rational expressions that need to be added or subtracted. The LCD is the smallest expression or product that all denominators can divide into evenly. This process often involves:

• Identifying unique factors from each denominator, ensuring that each is represented at its highest power.
• Combining these factors to form a single denominator expression.

For instance, in the given expressions, the denominators feature expressions like \(12x^3\), \(4(x+3)(x-1)\), and \(16(x-1)^2\).

To calculate the LCD:

• Consider each factor's highest powers. You have \(2^4\), \(3\), \(x^3\), \((x+3)\), and \((x-1)^2\).
• Multiplying these together gives you: \(\text{LCD} = 2^4 \cdot 3 \cdot x^3 \cdot (x+3)(x-1)\).

This LCD then allows each fraction to be rewritten with the same denominator, paving the way for straightforward addition or subtraction.

Simplifying Expressions

Simplifying expressions ensures that they are in their most reduced form, which makes further mathematical operations more manageable. This includes rewriting expressions with common denominators and combining terms to reduce complexity.

In the example, after rewriting each rational expression with the LCD, you can merge numerators. This involves distributing any multipliers and combining like terms to minimize the expression.

For the numerators in:

• \(4(x^2+3x-2)\)
• \(24x^3(x+1)\)
• \(- 3x^3(x+3)^2\)

Each needs expansion and simplification.

You'll distribute and combine terms like \(x^4, x^3, x^2\), etc., into a single rational expression. Ultimately, this process yields a simplified fraction that represents the same value as the original complex expression.

This strategy not only resolves the exercise but is applicable widely in algebra, making seemingly arduous expressions manageable and concise.