Problem 85 - Elementary Algebra

Question

For the following problems, convert the given rational expressions to rational expressions having the same denominators. $$ \frac{x-5}{x^{2}-9 x+20}, \frac{4}{x^{2}-3 x-10} $$

Step-by-Step Solution

Answer

Question: Convert the given rational expressions to have the same denominator: $$ \frac{x-5}{x^{2}-9 x+20} \text{ and } \frac{4}{x^{2}-3 x-10} $$ Answer: The rational expressions with the same denominator are: $$ \frac{(x-5)(x+2)}{(x-4)(x-5)(x+2)}, \frac{4(x-4)}{(x-4)(x-5)(x+2)} $$

Step 1: Factor the denominators

First, we need to factor the denominators of the given rational expressions: Denominator 1: \(x^{2}-9x+20\) Factoring this polynomial, we get: \((x-4)(x-5)\) Denominator 2: \(x^{2}-3x-10\) Factoring this polynomial, we get: \((x-5)(x+2)\)

Step 2: Calculate the Least Common Denominator (LCD)

The least common denominator is the smallest expression that is a multiple of both denominators. In this case, the LCD should have the factors \((x-4)(x-5)(x+2)\), as these factors cover all unique factors found in the original denominators: LCD = \((x-4)(x-5)(x+2)\)

Step 3: Rewrite the rational expressions with the LCD

Now, rewrite each rational expression with the LCD as the new denominator: Expression 1: $$ \begin{aligned} \frac{x-5}{(x-4)(x-5)} &= \frac{(x-5)}{(x-4)(x-5)} \cdot \frac{(x+2)}{(x+2)} \\ &= \frac{(x-5)(x+2)}{(x-4)(x-5)(x+2)} \end{aligned} $$ Expression 2: $$ \begin{aligned} \frac{4}{(x-5)(x+2)} &= \frac{4}{(x-5)(x+2)} \cdot \frac{(x-4)}{(x-4)} \\ &= \frac{4(x-4)}{(x-4)(x-5)(x+2)} \end{aligned} $$

Step 4: Final Result and Conclusion

The rational expressions converted to have the same denominator are: $$ \frac{(x-5)(x+2)}{(x-4)(x-5)(x+2)}, \frac{4(x-4)}{(x-4)(x-5)(x+2)} $$ Now, both rational expressions have the common denominator \((x-4)(x-5)(x+2)\).

Key Concepts

Factoring Polynomials

Factoring polynomials means breaking down a complex expression into simpler factors that, when multiplied together, give back the original expression. Understanding how to factor polynomials is crucial in algebra because it simplifies many operations, such as simplifying fractions and finding roots.
Imagine you have a polynomial like \( x^2 - 9x + 20 \). To factor it, you're looking for two numbers that multiply to the constant term (here, 20) and add up to the linear coefficient (here, -9). The numbers are -4 and -5. Therefore, the expression factors to \((x-4)(x-5)\).
When factoring, always look out for common patterns like difference of squares, perfect square trinomials, or simply find common terms. Sometimes, polynomials need rearranging or trial and error to identify these patterns. Having a wide range of strategies at your disposal will make factoring much more manageable.

Least Common Denominator

The Least Common Denominator (LCD) is the smallest expression that can be used as a common denominator for two or more fractions. This is particularly useful when you need to compare, add, or subtract rational expressions.
To find the LCD for rational expressions, factor each denominator first. Then, take each unique factor from the set of denominators and multiply them together. For example, consider the denominators \((x-4)(x-5)\) and \((x-5)(x+2)\). The LCD should include all unique factors, which means combining these to make \((x-4)(x-5)(x+2)\).
By using the LCD, you're ensuring that you have a standard to simplify and combine expressions. This commonality allows for cleaner manipulation of algebraic fractions and prepares the groundwork for further algebraic operations with ease.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions, such as polynomials. They behave much like numerical fractions, but you need to apply algebraic operations such as factoring and simplification.
When dealing with algebraic fractions, it's often necessary to rewrite them to have common denominators, especially if you're adding or subtracting. This means factoring the denominators, finding the least common denominator, and then rewriting the fractions.
For example, if you start with \(\frac{x-5}{(x-4)(x-5)}\) and \(\frac{4}{(x-5)(x+2)}\), you rewrite these so they both have \((x-4)(x-5)(x+2)\) as the denominator. This involves multiplying the fractions by forms of 1 using missing factors: \(\frac{(x+2)}{(x+2)}\) for the first expression and \(\frac{(x-4)}{(x-4)}\) for the second. By adjusting these algebraic fractions to have common denominators, you align them for easy calculation and comparison.