Problem 20 - Elementary Algebra

Question

Change the given rational expressions into rational expressions with the same denominators. $$ \frac{10 x}{x^{2}+8 x+16}, \frac{5 x}{x^{2}-16} $$

Step-by-Step Solution

Answer

Question: Rewrite the given rational expressions with the same denominator: $\frac{10x}{x^2+8x+16}$ and $\frac{5x}{x^2-16}$. Answer: The rational expressions with the same denominator are: $\frac{10x(x-4)}{(x+4)^2(x-4)}$ and $\frac{5x(x+4)}{(x+4)^2(x-4)}$.

Step 1: Factor the denominators

Factor the denominators of both rational expressions: $$ (x^2 + 8x + 16) = (x+4)^2 \\ (x^2 - 16) = (x+4)(x-4) $$

Step 2: Determine the LCD

Find the least common denominator (LCD) by considering which factors must be included in both denominators. The LCD must have the factors \((x+4)^2\) and \((x-4)\). So, the LCD is \((x+4)^2(x-4)\).

Step 3: Rewrite the expressions with the same denominator

Rewrite both rational expressions with their common denominator \((x+4)^2(x-4)\). Multiply both the numerator and denominator of the first rational expression by \((x-4)\): $$ \frac{10x}{(x+4)^2} \cdot \frac{(x-4)}{(x-4)} = \frac{10x(x-4)}{(x+4)^2(x-4)} $$ Multiply both the numerator and denominator of the second rational expression by \((x+4)\): $$ \frac{5x}{(x+4)(x-4)} \cdot \frac{(x+4)}{(x+4)} = \frac{5x(x+4)}{(x+4)^2(x-4)} $$ The two rational expressions with the same denominator are: $$ \frac{10x(x-4)}{(x+4)^2(x-4)}, \frac{5x(x+4)}{(x+4)^2(x-4)} $$

Key Concepts

Least Common Denominator

A common challenge when working with rational expressions is finding the least common denominator (LCD). The LCD is the smallest expression that includes all the factors present in the denominators of each rational expression. To find it, you need to consider each denominator and factor it completely. For example, in our exercise, we factored \(x^2 + 8x + 16\) as \((x+4)^2\) and \(x^2 - 16\) as \((x+4)(x-4)\). From these, the LCD is determined by taking the highest power of each factor present: \((x+4)^2\) and \((x-4)\). Thus, our LCD is \((x+4)^2(x-4)\).

Once you have identified the LCD, the next step is to rewrite each rational expression so that their denominators match the LCD. This ensures they can be easily added or subtracted if necessary.

Factoring

Factoring is a crucial skill needed when dealing with polynomial expressions, like those found in rational expressions. It involves breaking down complex polynomials into simpler factors. In our example, the process starts with recognizing patterns or utilizing methods such as the difference of squares, trinomial factoring, or grouping.

For \(x^2 + 8x + 16\), you notice it's a perfect square trinomial and factor it as \((x+4)^2\). For \(x^2 - 16\), recognizing the difference of squares leads to \((x+4)(x-4)\).

• Always look for common factors first across all terms.
• Check if there are any special patterns that simplify the expression.
• For trinomials, try to rewrite as squares or factors of two binomials if possible.

Factorizing allows us to see the component parts of the expression, which is essential when finding the least common denominator.

Polynomial Expressions

Polynomial expressions form the basis of many algebraic structures, including rational expressions. These are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. The degree of the polynomial is the highest power of the variable within the expression.

Working with polynomials implies understanding their structure and how they behave arithmetically—whether through addition, subtraction, multiplication, or division (factoring included). For instance, recognizing that \(x^2 + 8x + 16\) is a quadratic polynomial helps identify it for factoring as a trinomial. Similarly, \(x^2 - 16\) is a difference of squares, a type of polynomial that can be easily factored.

This manipulation of polynomial expressions is key in simplifying and calculating with rational expressions, especially when finding common denominators and simplifying the terms.