Problem 50 - Algebra and Trigonometry

Question

\(35-54\) . Perform the addition or subtraction and simplify. $$ \frac{x}{x^{2}+x-2}-\frac{2}{x^{2}-5 x+4} $$

Step-by-Step Solution

Answer

\(\frac{x^2 - 6x - 4}{(x + 2)(x - 1)(x - 4)}\) is the simplified expression.

Step 1: Factor the Denominators

To simplify the expression, we first need to factor each of the denominators. The first denominator is \(x^2 + x - 2\). We factor it by finding two numbers that multiply to \(-2\) and add to \(1\). These numbers are \(2\) and \(-1\), so \(x^2 + x - 2 = (x + 2)(x - 1)\). The second denominator is \(x^2 - 5x + 4\). We factor it by finding two numbers that multiply to \(4\) and add to \(-5\). These are \(-4\) and \(-1\), so \(x^2 - 5x + 4 = (x - 4)(x - 1)\).

Step 2: Identify the Common Denominator

Both denominators need to be combined into a common denominator to subtract the fractions. The common denominator will be the product of all distinct factors from both denominators, which is \((x + 2)(x - 1)(x - 4)\).

Step 3: Adjust Each Fraction to the Common Denominator

Adjust each fraction to have the common denominator. For \(\frac{x}{x^2 + x - 2}\), we need to multiply both the numerator and denominator by \((x - 4)\). This gives \(\frac{x(x - 4)}{(x + 2)(x - 1)(x - 4)}\). For \(\frac{2}{x^2 - 5x + 4}\), we need to multiply both the numerator and denominator by \((x + 2)\), resulting in \(\frac{2(x + 2)}{(x + 2)(x - 1)(x - 4)}\).

Step 4: Subtract the Fractions

Now that both fractions have a common denominator, we can subtract the numerators: \(\frac{x(x - 4) - 2(x + 2)}{(x + 2)(x - 1)(x - 4)}\). Simplify the numerator to \(x^2 - 4x - 2x - 4 = x^2 - 6x - 4\). So the expression becomes \(\frac{x^2 - 6x - 4}{(x + 2)(x - 1)(x - 4)}\).

Step 5: Simplify the Expression

Finally, check if the numerator \(x^2 - 6x - 4\) can be factored to simplify further. However, \(x^2 - 6x - 4\) does not factor neatly with integer coefficients, thus the final simplified expression is \(\frac{x^2 - 6x - 4}{(x + 2)(x - 1)(x - 4)}\).

Key Concepts

Factoring

Factoring in mathematics is the process of breaking down a complex expression into simpler components, known as factors, that when multiplied together give the original expression. It is a critical concept, especially when dealing with polynomial expressions. In the context of rational expressions, like the ones in our exercise, factoring helps in simplifying the algebraic fractions.
To factor a quadratic expression such as \(x^2 + x - 2\), we look for two numbers that multiply to give us the constant term (-2) and add up to the coefficient of the middle term (1). In this case, the numbers 2 and -1 do the job. Therefore, \(x^2 + x - 2\) can be expressed as \((x + 2)(x - 1)\).
For \(x^2 - 5x + 4\), we seek numbers that multiply to 4 and sum to -5, which are -4 and -1. Hence, we factor it into \((x - 4)(x - 1)\).
Factoring is useful because it simplifies expressions by breaking them into smaller, more manageable pieces. This factored form is then used later when determining the common denominator, an essential step in operations with rational expressions.

Common Denominator

Finding a common denominator is essential when performing addition or subtraction with rational expressions. The common denominator is the shared denominator across all terms that allows you to perform these operations. In our exercise, we start by factoring the individual denominators which are \((x + 2)(x - 1)\) for the first fraction and \((x - 4)(x - 1)\) for the second.
The next step is to identify all distinct factors from these expressions. A common denominator will include every distinct factor, ensuring it encompasses both denominators. Hence, the common denominator for our fractions becomes \((x + 2)(x - 1)(x - 4)\).
This process converts the denominators into a suitable form that allows the subtraction of the fractions. Ensuring a common denominator is critical because it standardizes the fractions, making the numerators the only part needing adjustment by multiplication.

Simplifying Fractions

Simplifying fractions, especially in rational expressions, involves reducing the expression to its simplest form. After finding a common denominator and combining the numerators, we need to evaluate if the resulting expression can be further reduced.
In this exercise, once we have a single numerator from the expression \(x(x - 4) - 2(x + 2)\), we simplify it to get \(x^2 - 6x - 4\). At this stage, checking for further factorization of the numerator is crucial. However, \(x^2 - 6x - 4\) doesn't factor cleanly with integer coefficients, indicating that the result is already in its simplest form.
When simplifying fractions, it's important to remember:

• Factoring both the numerator and the denominator completely.
• Canceling any common factors between the numerator and the denominator if possible.

The absence of further factors in the numerator and denominator ensures the expression is fully simplified.