Question

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

Short answer

\(\frac{x^{2} + 5}{x^{2}(x + 1)(x - 1)}\)

Step-by-step solution

Simplify each term

First, simplify the denominators of each fraction separately. The first term, \(\frac{1}{x}\), is already in its simplest form.

Factor the second denominator

The second term is \(\frac{2}{x^{2}+x}\). Factor the denominator: \(x(x + 1)\). So, it becomes \(\frac{2}{x(x + 1)}\).

Factor the third denominator

The third term is \(\frac{3}{x^{3}-x^{2}}\). Factor the denominator: \(x^{2}(x - 1)\). So, it becomes \(\frac{3}{x^{2}(x - 1)}\).

Determine the common denominator

Determine the least common multiple (LCM) of the denominators: \(x\), \(x(x+1)\), and \(x^{2}(x-1)\). The common denominator is \(x^{2}(x + 1)(x - 1)\).

Rewrite each fraction with the common denominator

Multiply each term so that all have the common denominator \(x^{2}(x + 1)(x - 1)\).\(\frac{1}{x} = \frac{x(x + 1)(x - 1)}{x^{2}(x + 1)(x - 1)} = \frac{x^{2} - x}{x^{2}(x + 1)(x - 1)}\)\(\frac{2}{x(x + 1)} = \frac{2x(x - 1)}{x^{2}(x + 1)(x - 1)} = \frac{2x - 2}{x^{2}(x + 1)(x - 1)}\)\(\frac{3}{x^{2}(x - 1)} = \frac{3(x + 1)}{x^{2}(x + 1)(x - 1)} = \frac{3x + 3}{x^{2}(x + 1)(x - 1)}\)

Combine the fractions

Combine the fractions: \(\frac{x^{2} - x}{x^{2}(x + 1)(x - 1)} - \frac{2x - 2}{x^{2}(x + 1)(x - 1)} + \frac{3x + 3}{x^{2}(x + 1)(x - 1)}\)Combine the numerators: \(\frac{x^{2} - x - 2x + 2 + 3x + 3}{x^{2}(x + 1)(x - 1)}\)

Simplify the numerator

Simplify the numerator: \(x^{2} - x - 2x + 2 + 3x + 3 = x^{2} + (-x - 2x + 3x) + (2 + 3) = x^{2} + 2 + 5 = x^{2} + 5\)Then our expression becomes: \(\frac{x^{2} + 5}{x^{2}(x + 1)(x - 1)}\)

Factor the simplified numerator and denominator

Determine if the numerator or denominator can be further factored. Since they can't be factored further, the final expression remains in its simplified, factored form.

Key concepts

factoring polynomials

Factoring polynomials is all about breaking down a complex polynomial into simpler factors, which are polynomials of lower degrees.
The factors when multiplied together give back the original polynomial. This is a powerful tool, especially when working with fractions in algebra.
In the given exercise:

• The denominator of the second term, \(\frac{2}{x^{2}+x}\) is factored as \(x(x + 1)\).
• The denominator of the third term, \(\frac{3}{x^{3}-x^{2}}\) is factored as \(x^{2}(x - 1)\).

Factoring polynomials helps in simplifying and managing the expressions effectively.
The key is to look for common terms or patterns in the polynomial.
Remembering factorization formulas such as difference of squares (\(a^2 - b^2 = (a - b)(a + b)\)), or recognizing the greatest common factor will make your efforts fruitful.

least common multiple

The Least Common Multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.
When adding or subtracting fractions, finding the LCM of the denominators is crucial.
In algebra, this concept extends to polynomial denominators as well.
In the exercise, you have the denominators \x\, \x(x + 1)\, and \x^{2}(x - 1)\.
The LCM of these denominators is \(x^{2}(x + 1)(x - 1)\).

• For \x\: We need to consider it as \(x^{2}(x - 1)(x + 1)\) to include all needed factors.
• For \(x(x + 1)\): Similarly, convert \(x(x + 1)\) to \(x^{2}(x + 1)(x - 1)\) for inclusion.
• For \(x^{2}(x - 1)\): It directly fits within \(x^{2}(x + 1)(x - 1)\).

Finding the LCM standardizes the denominators and allows you to combine the fractions effortlessly.

simplifying expressions

Simplifying expressions involves reducing a complex expression into its simplest form.
It often includes combining like terms, reducing fractions, and factoring.
In this particular exercise, simplifying includes:

• Combining like terms in the numerator of \(\frac{x^{2} - x - 2x + 2 + 3x + 3}{x^{2}(x + 1)(x - 1)}\) to get \(x^{2} + 5\).
• Ensuring the expression is left in the most reduced, factored form: \(\frac{x^{2} + 5}{x^{2}(x + 1)(x - 1)}\).

Each step in the exercise builds on the previous one,
making it essential to follow the logical progression.
Mastering simplification allows for a clearer, more understandable solution and prepares you for more complex problems.