Question

Perform the indicated operations and simplify the result. Leave your answer in factored form. $$ \frac{\frac{2 x+5}{x}-\frac{x}{x-3}}{\frac{x^{2}}{x-3}-\frac{(x+1)^{2}}{x+3}} $$

Short answer

\( \frac{(x+5)(x+3)}{x} \)

Step-by-step solution

Simplify the Numerator

Start by simplifying the numerator: \( \frac{2x+5}{x} - \frac{x}{x-3} \). To combine these fractions, first find a common denominator. The common denominator is \( x(x-3) \). Rewrite each fraction with the common denominator:\( \frac{(2x+5)(x-3) - x^2}{x(x-3)} \).

Expand the Numerator

Expand the terms in the numerator:\( (2x+5)(x-3) = 2x^2 - 6x + 5x - 15 = 2x^2 - x - 15 \).So, the combined numerator becomes:\( \frac{2x^2 - x - 15 - x^2}{x(x-3)} = \frac{x^2 - x - 15}{x(x-3)} \).

Simplify the Denominator

Next, simplify the denominator: \( \frac{x^2}{x-3} - \frac{(x+1)^2}{x+3} \). Find a common denominator, which is \( (x-3)(x+3) \). Rewrite each fraction with the common denominator:\( \frac{x^2(x+3) - (x+1)^2(x-3)}{(x-3)(x+3)} \).

Expand the Denominator

Expand the terms in the denominator:\[ x^2(x+3) = x^3 + 3x^2 \]\[ (x+1)^2(x-3) = (x^2 + 2x + 1)(x-3) = x^3 - 3x^2 + 2x^2 - 6x + x - 3 = x^3 - x^2 - 5x - 3 \]So, the combined denominator becomes:\( \frac{x^3 + 3x^2 - (x^3 - x^2 - 5x - 3)}{(x-3)(x+3)} = \frac{x^2 + 5x + 3}{(x-3)(x+3)} \).

Form the Single Fraction

Combine the simplified numerator and simplified denominator:\( \frac{\frac{x^2 - x - 15}{x(x-3)}}{\frac{x^2 + 5x + 3}{(x-3)(x+3)}} = \frac{x^2 - x - 15}{x(x-3)} \times \frac{(x-3)(x+3)}{x^2 + 5x + 3} \).

Cancel Common Factors

Cancel out the common factors in the numerator and the denominator:\( \frac{(x-3)}{(x-3)} \text{ cancels out, so we get} \frac{(x^2 - x - 15)(x+3)}{x(x^2 + 5x + 3)} \).

Factor and Simplify

Factor the remaining terms where possible:\( x^2 - x - 15 = (x - 3)(x + 5) \)\( x^2 + 5x + 3 \text{ is already simplified} \)So we have:\( \frac{(x-3)(x+5)(x+3)}{x(x^2 + 5x + 3)} \).

Final Factored Form

The final step is to express the result in factored form:\( \frac{(x+5)(x+3)}{x} \).

Key concepts

fraction_operations

Understanding fraction operations is important in algebra. In this exercise, you need to combine the fractions in the numerator and the fractions in the denominator before dividing them.

• First, identify the fractions involved. Here, they are \ \( \frac{2x+5}{x} \ \) and \ \( \frac{x}{x-3} \ \) in the numerator and \ \( \frac{x^2}{x-3} \ \) and \ \( \frac{(x+1)^2}{x+3} \ \) in the denominator.


• Next, find a common denominator to combine the fractions. For the numerator, it is \ \( x(x-3) \ \), and for the denominator, it is \ \( (x-3)(x+3) \ \).


• Rewrite each fraction with the common denominator, and then combine the numerators and denominators. For example, the numerator becomes \ \( \frac{(2x+5)(x-3) - x^2}{x(x-3)} \ \) after combining.

Finally, understanding how to manipulate these fractions by finding common denominators and combining them is crucial for later steps in simplifying and factoring.

factoring_polynomials

Factoring polynomials is a key algebraic skill needed to simplify expressions. In this exercise, you factor the quadratic expressions.

• For example, consider the numerator: \ \( 2x^2 - x - 15 \ \). This polynomial needs to be factored into a product of two binomials.


• Start by identifying pairs of numbers that multiply to give the constant term (-15) and add to give the coefficient of the middle term (-1).


• The factors of \ \( 2x^2 - x - 15 \ \) are \ \( (2x+5)(x-3) \ \). These factors will be crucial later when simplifying the overall fraction.

The same approach applies to other polynomials in the exercise, such as looking for common factors or using the quadratic formula if necessary.

simplifying_expressions

Simplifying expressions involves reducing them to their most basic form. In this exercise, after factoring, you simplify by canceling common terms.

• After rewriting the fractions with common denominators, combine like terms in the numerator and denominator. For instance, combining terms in the numerator of step 2 results in: \ \( \frac{x^2 - x - 15}{x(x-3)} \ \).


• Next, look for factors that can be canceled. For example, in step 6, you cancel \ \( (x - 3) \ \) terms in the numerator and denominator.


• Ensure what remains is in the simplest form possible. At the end of these operations, you'll end up with simplified expressions like \ \( \frac{(x + 5)(x + 3)}{x} \ \).

Simplifying expressions helps make complex problems more manageable and easier to interpret.

common_denominator

Finding a common denominator is essential in combining fractions. It ensures that fractions can be added or subtracted correctly.

• First, identify the different denominators in your fractions. For example, in step 1, they are \ \( x \ \) and \ \( x-3 \ \).


• Determine the least common multiple (LCM) of these denominators. The LCM of \ \( x \ \) and \ \( x-3 \ \) is \ \( x(x-3) \ \).


• Rewrite each fraction so that they each share this common denominator. For the numerator, make: \ \( \frac{(2x+5)(x-3)}{x(x-3)} - \frac{x(x)}{x(x-3)} \ \).

Similarly, find common denominators for the fractions in the denominator of the original expression to streamline the process of combining fractions.
This step ensures fractions are in a comparable form, which is critical for any operations on them.