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Magazine Chapter 7: Problem 42
Add or subtract. Write answer in lowest terms. \(\frac{2 x}{x-1}+\frac{-4}{x^{2}-1}\)
Short Answer
Expert verified
\[\frac{2(x+2)}{x+1}\]
Step by step solution
01
- Factor the denominators
Identify and factor the denominators. The second denominator is a difference of squares: \[x^2 - 1 = (x - 1)(x + 1)\]
02
- Find the common denominator
The first fraction's denominator is \(x - 1\). To make a common denominator, multiply the numerator and denominator of the first fraction by \(x + 1\), which is the missing factor from the second denominator: \[\frac{2x(x + 1)}{(x - 1)(x + 1)}\]
03
- Rewrite fractions with the common denominator
Rewrite each fraction with the common denominator: \[\frac{2x(x+1)}{(x-1)(x+1)} + \frac{-4}{(x-1)(x+1)}\]
04
- Combine the numerators
Combine the fractions by adding the numerators: \[\frac{2x(x+1) - 4}{(x-1)(x+1)} = \frac{2x^2 + 2x - 4}{(x-1)(x+1)}\]
05
- Simplify the numerator
Factor the numerator if possible. Notice that the numerator \(2x^2 + 2x - 4\) can be factored as \(2(x^2 + x - 2)\): \[\frac{2(x^2 + x - 2)}{(x-1)(x+1)}\]
06
- Factor the trinomial
Factor the trinomial \(x^2 + x - 2\). It factors as \((x + 2)(x - 1)\): \[\frac{2(x + 2)(x - 1)}{(x-1)(x+1)}\]
07
- Cancel common factors
Cancel the common factor \(x - 1\) from the numerator and the denominator: \[\frac{2(x+2)}{x+1}\]
08
- Write the final simplified form
The simplified form of the expression is: \[\frac{2(x+2)}{x+1}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Denominators
To add or subtract rational expressions, they must have a common denominator. A common denominator allows us to combine the fractions into one. This is much like finding a common denominator for regular fractions. Here, we need both expressions to have the same bottom part. For example, in the problem \(\frac{2x}{x-1} + \frac{-4}{x^2-1}\), the denominators are initially different. The first is \(x-1\), but the second can be factored into \( (x-1)(x+1) \). By making the denominators the same, we can easily combine the fractions.
Factoring Trinomials
Factoring trinomials is a crucial skill in algebra. A trinomial is a type of polynomial with three terms. When you need to simplify or solve expressions, factoring can break down complex polynomials into simpler binomials or monomials. For instance, in Step 6 of our solution, the numerator \(x^2 + x - 2\) is factored into \( (x + 2)(x - 1) \). This is very useful for simplifying and solving algebraic fractions. To factor a trinomial, look for two numbers that multiply to the last term and add up to the middle term.
Simplifying Algebraic Fractions
Simplifying algebraic fractions involves several steps: factoring, finding a common denominator, and reducing the expression. In our problem, after factoring the denominators and rewriting the fractions with a common denominator, we combine and simplify the expression. For example: \(\frac{2x^2 + 2x - 4}{(x-1)(x+1)} \) can be further simplified to \(\frac{2(x^2 + x - 2)}{(x-1)(x+1)} \), and then to \(\frac{2(x+2)(x-1)}{(x-1)(x+1)} \). Lastly, by canceling out common factors, we get \(\frac{2(x+2)}{x+1} \). Simplification makes the expression much easier to understand and work with.
Difference of Squares
The difference of squares is a tool used in factoring specific types of expressions. It occurs when you have two squared terms separated by a subtraction sign, such as \( x^2 - 1 \). This expression can be factored into \( (x - 1)(x + 1) \). Recognizing and factoring differences of squares help simplify algebraic fractions and make it easier to find common denominators. In our example, recognizing \( x^2 - 1 \) as \( (x-1)(x+1) \) was essential to progress with the solution.
