Question

Perform the indicated operations and simplify the result. Leave your answer in factored form. $$ \frac{4 x}{x^{2}-4}-\frac{2}{x^{2}+x-6} $$

Short answer

\(\frac{4(x^2 + 2.5x - 1)}{(x - 2)(x + 2)(x + 3)}\)

Step-by-step solution

- Factor the denominators

First, factor the denominators of each fraction.For \(x^2 - 4\), factor it as a difference of squares:\[x^2 - 4 = (x - 2)(x + 2)\]For \(x^2 + x - 6\), factor it as a product of binomials:\[x^2 + x - 6 = (x + 3)(x - 2)\]

- Rewrite the fractions with factored denominators

Rewrite each fraction using the factorized forms of the denominators:\[\frac{4x}{(x - 2)(x + 2)} - \frac{2}{(x + 3)(x - 2)}\]

- Find a common denominator

Determine the common denominator by combining the factors from both denominators:\[(x - 2)(x + 2)(x + 3)\]

- Rewrite each fraction with the common denominator

Rewrite each fraction so they both have the common denominator:\[\frac{4x(x + 3)}{(x - 2)(x + 2)(x + 3)} - \frac{2(x + 2)}{(x - 2)(x + 3)(x + 2)}\]

- Combine the fractions

Combine the two fractions over the common denominator:\[\frac{4x(x + 3) - 2(x + 2)}{(x - 2)(x + 2)(x + 3)}\]

- Simplify the numerator

Simplify the expression in the numerator:Expand and combine like terms:\[4x(x + 3) - 2(x + 2) = 4x^2 + 12x - 2x - 4 = 4x^2 + 10x - 4\]Factor the simplified numerator:\[4(x^2 + 2.5x - 1)\]

- Write the final expression

Combine the simplified numerator and the common denominator:\[\frac{4(x^2 + 2.5x - 1)}{(x - 2)(x + 2)(x + 3)}\]

- Factor the numerator further if possible (optional)

If possible, factor the quadratic in the numerator further, otherwise, leave as it is.

Key concepts

Factoring

In algebra, factoring refers to the process of breaking down an expression into simpler components (called factors) that, when multiplied together, give the original expression. For instance, the expression \(x^2 - 4\) can be factored as \((x - 2)(x + 2)\), because \( (x - 2) \times (x + 2) = x^2 - 4 \). Factoring is especially useful when working with rational expressions and can simplify the process of adding, subtracting, multiplying, or dividing them.
In our exercise, we factor two different quadratic expressions: \(x^2 - 4\) and \(x^2 + x - 6\). These factorizations help us to identify a common denominator for the given rational expressions.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. To perform operations like addition and subtraction on fractions, they must have the same denominator.
For example, consider the fractions \( \frac{4x}{(x - 2)(x + 2)} \) and \( \frac{2}{(x + 3)(x - 2)}\) from our problem. Here, the common denominator can be identified by combining all the distinct factors from both denominators, which gives us \( (x - 2)(x + 2)(x + 3) \).
Once we have the common denominator, we rewrite each fraction to have this shared base, making it easier to combine them.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. These can be simplified, added, subtracted, multiplied, or divided just like numerical fractions, but they often require factoring and finding common denominators.
For instance, in the given problem, \( \frac{4x}{x^2-4} - \frac{2}{x^2+x-6} \) involves two rational expressions with polynomials in their denominators. By factoring these polynomials and finding a common denominator, we simplify the task of subtracting the fractions.
Proper handling of rational expressions is crucial in algebra and is encountered frequently in higher mathematics.

Simplifying Fractions

Simplifying fractions makes them easier to work with by expressing them in their simplest form. This is done by dividing the numerator and the denominator by their greatest common factor (GCF).
In the problem, the expression \( \frac{4x(x + 3) - 2(x + 2)}{(x - 2)(x + 2)(x + 3)} \) is simplified by first combining the numerators and then simplifying:
* Expand the numerator: \( 4x(x + 3) - 2(x + 2) = 4x^2 + 12x - 2x - 4 \)
* Combine like terms: \( 4x^2 + 10x - 4 \)
* If possible, factor the resultant polynomial in the numerator further

The final simplified form of the fraction will have the simplified numerator over the common denominator. Simplifying fractions ensures expressions are as concise as possible, reducing errors and improving clarity in calculations. Substituting any common factors can sometimes reveal further simplification opportunities.