Question

Perform the indicated operations and simplify the result. Leave your answer in factored form. $$ \frac{x}{x^{2}-7 x+6}-\frac{x}{x^{2}-2 x-24} $$

Short answer

The simplified result is \( \frac{5x}{(x-1)(x-6)(x+4)} \).

Step-by-step solution

- Factor the Denominators

First, factor the denominators of the fractions. The denominators are quadratic expressions. For the first fraction: \[ x^{2} - 7x + 6 \] Factoring, we get: \[ (x - 1)(x - 6) \] For the second fraction: \[ x^{2} - 2x - 24 \] Factoring, we get: \[ (x - 6)(x + 4) \]

- Rewrite the Fractions

Rewrite each fraction with their factored denominators: \[ \frac{x}{(x-1)(x-6)} - \frac{x}{(x-6)(x+4)} \]

- Find a Common Denominator

Find a common denominator, which is the product of all distinct factors in the denominators. In this case, the common denominator is: \[ (x - 1)(x - 6)(x + 4) \]

- Rewrite the Fractions with the Common Denominator

Rewrite each fraction so they have the common denominator: For the first fraction: \[ \frac{x}{(x-1)(x-6)} \times \frac{(x+4)}{(x+4)} = \frac{x(x+4)}{(x-1)(x-6)(x+4)} \] For the second fraction: \[ \frac{x}{(x-6)(x+4)} \times \frac{(x-1)}{(x-1)} = \frac{x(x-1)}{(x-6)(x+4)(x-1)} \]

- Combine the Fractions

Combine the two fractions over the common denominator: \[ \frac{x(x+4) - x(x-1)}{(x-1)(x-6)(x+4)} \]

- Simplify the Numerator

Simplify the numerator: \[ x(x + 4) - x(x - 1) = x^{2} + 4x - x^{2} + x = 5x \]

- Write the Final Expression

Substitute the simplified numerator back into the combined fraction: \[ \frac{5x}{(x-1)(x-6)(x+4)} \]

Key concepts

Factoring Quadratics

To handle algebraic fractions, we first need to factor the quadratic expressions in the denominators. Quadratic expressions are of the form \( ax^{2} + bx + c \). Factoring quadratics involves finding two binomials whose product is the quadratic. For instance, in our exercise, we have \( x^{2} - 7x + 6 \). The factors of this quadratic expression are \( (x - 1)(x - 6) \). What we did here is find two numbers that multiply to 6 (the constant term) and add up to -7 (the coefficient of the linear term). Similarly, for \( x^{2} - 2x - 24 \), the factors are \( (x - 6)(x + 4) \). breaking down these steps helps ensure that the subsequent operations remain correct and simpler to handle.

Finding a Common Denominator

To perform operations like subtraction on fractions, we need a common denominator. This is crucial because we can only subtract or add fractions directly if they have the same denominator. From our factored quadratic denominators – \( (x - 1)(x - 6) \) and \( (x - 6)(x + 4) \) – the common denominator will be their combined factors: \( (x - 1)(x - 6)(x + 4) \). It includes all distinct factors from both denominators. This allows us to rewrite each fraction with this common denominator before performing any operations.

Fraction Simplification

After finding the common denominator, each fraction must be rewritten so they share this common denominator. This involves multiplying the numerator and the denominator of each fraction by whatever term is missing from their individual denominators. For the first fraction, \( \frac{x}{(x - 1)(x - 6)} \times \frac{(x + 4)}{(x + 4)} \) results in \( \frac{x(x + 4)}{(x - 1)(x - 6)(x + 4)} \). For the second fraction, \( \frac{x}{(x - 6)(x + 4)} \times \frac{(x - 1)}{(x - 1)} \) gives \( \frac{x(x - 1)}{(x - 6)(x + 4)(x - 1)} \). Making the denominators uniform allows for direct subtraction or addition of the fractions.

Subtraction of Fractions

Subtracting fractions with a common denominator involves subtracting their numerators and keeping the denominator the same. For example: \( \frac{x(x + 4)}{(x - 1)(x - 6)(x + 4)} - \frac{x(x - 1)}{(x - 6)(x + 4)(x - 1)} \). We subtract their numerators: \( x(x + 4) - x(x - 1) \). Written out, it is \( x^{2} + 4x - x^{2} + x \). In the numerator, the \( x^{2} \) terms cancel each other, leaving \( 5x \). Thus, the fraction simplifies to \( \frac{5x}{(x - 1)(x - 6)(x + 4)} \).

Algebraic Expression Simplification

Finally, simplifying an algebraic expression requires combining like terms and reducing the overall expression to lowest terms. This can involve canceling out terms in the numerator and the denominator. In the given exercise, after subtracting the numerators, we got \( 5x \) in the numerator. We insert this back into the fraction giving us \( \frac{5x}{(x - 1)(x - 6)(x + 4)} \). Since there are no common factors to cancel out in the numerator and the denominator, this is the fully simplified form. Ensuring every step is properly simplified is key to mastering algebraic fractions.