Problem 49 - Algebra 1

Question

When you add rational expressions, you may need to factor a trinomial to find the LCD. Study the sample below. Then simplify the expressions in Exercises 46–49. $$\text { Sample: } \frac{2 x}{x^{2}-1}+\frac{3}{x^{2}+x-2}=\frac{2 x}{(x+1)(x-1)}+\frac{3}{(x-1)(x+2)}$$ The LCD is \((x+1)(x-1)(x+2)\) Note: If you just used \(\left(x^{2}-1\right)\left(x^{2}+x-2\right)\) as the common denominator, the factor \((x-1)\) would be included twice. $$\frac{5 x-1}{2 x^{2}-7 x-15}-\frac{-3 x+4}{2 x^{2}+5 x+3}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{8x^2-18x+19}{(x-1)(2x+3)(x+5)}\).

Step 1: Factor the Denominators

Begin by factoring the denominators: \(2 x^{2}-7 x-15 = 2x^2 -10x +3x -15 = 2x(x-5)+3(x-5)=(x-5)(2x+3)\) and \(2 x^{2}+5 x+3 = 2x^2 +2x +3x +3 =2x(x+1)+3(x+1)=(x+1)(2x+3)\). So the factored expressions become: \(\frac{5 x-1}{(x-5)(2x+3)} - \frac{-3 x+4}{(x+1)(2x+3)}\)

Step 2: Determine the LCD

The least common denominator (LCD) of the the two fractions is the product of the unique factors. In our case, the LCD is \((x-5)(2x+3)(x+1)\). Note that the factor \((2x+3)\) is not needed to be multiplied twice as the common factors are noted once in the LCD.

Step 3: Simplify the Expression

Rewrite the fractions with this LCD and simplify the numerators. Our expression looks like this: \(\frac{(5 x-1)(x+1)}{(x-5)(2x+3)(x+1)} - \frac{(-3 x+4)(x-5)}{(x+1)(2x+3)(x-5)} = \frac{(5x^2+x-1)-(-3x^2+19x-20)}{(x-1)(2x+3)(x+5)} = \frac{8x^2-18x+19}{(x-1)(2x+3)(x+5)}\)

Key Concepts

Least Common Denominator (LCD)

When dealing with rational expressions, especially in addition and subtraction, finding the Least Common Denominator (LCD) is a crucial step. The LCD is the smallest expression you can use as the common denominator of two or more fractions. It consists of all the different factors present in each of the denominators of those fractions.
For example, consider the sample sum of rational expressions given in the problem. The denominators were \((x^2-1)\) and \((x^2+x-2)\). By factoring them into \((x-1)(x+1)\) and \((x-1)(x+2)\) respectively, it's noted that \((x-1)\) is a common factor, so it only appears once in the LCD formula, resulting in \((x+1)(x-1)(x+2)\).

• Always factor every denominator completely.
• Include each factor the greatest number of times it appears in any one denominator.

This ensures that the LCD is truly the least, meaning it minimizes the complexity of the expressions you're working with.

Factoring Trinomials

Factoring trinomials is a technique used to simplify algebraic expressions, which can often be a part of identifying the least common denominator. A trinomial is typically a polynomial with three terms, and factoring it means expressing it as a product of two binomials.
Take a look at a step from our solution: the expression \(2x^2 - 7x - 15\) was rewritten as \((x-5)(2x+3)\). This process involves using methods such as:

• Finding two numbers that multiply to the constant term (the last term) and add up to the middle term (the coefficient of the \(x\) term).
• Applying splitting the middle term method.

By factoring the trinomial, it makes finding the LCD and simplifying the fractions more manageable. Factoring is key in converting complex expressions into simpler forms that are easier to manipulate.

Algebraic Fractions

Algebraic fractions, or rational expressions, are fractions where the numerator, the denominator, or both are algebraic expressions. This means they include variables, which can make them seem more challenging than ordinary fractions. However, the principles remain largely the same.
When dealing with algebraic fractions, here’s a brief strategy to follow:

• Simplify each fraction starting with the numerator and the denominator.
• Factor expressions as much as possible, like in the numerator \((5x^2 + x - 1)\) from our example.
• Identify common factors to find the LCD and rewrite each fraction over this common base.

Once all fractions have a common denominator, it becomes straightforward to perform operations like addition, subtraction, and comparing fractions. Handling algebraic fractions efficiently often hinges on a thorough understanding of factoring and simplification concepts.