Question

Perform the indicated operations and simplify the result. Leave your answer in factored form. $$ \frac{3 x}{x-1}-\frac{x-4}{x^{2}-2 x+1} $$

Short answer

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

Step-by-step solution

- Factor the Denominator

Start by factoring the denominator in the second fraction. Recognize that \(x^2 - 2x + 1\) can be factored as \( (x-1)^2\). So the second fraction becomes: \[ \frac{x-4}{(x-1)^2} \].

- Find a Common Denominator

Identify the common denominator for the two fractions. The first fraction has a denominator of \(x-1\), and the second fraction has a denominator of \((x-1)^2\). Therefore, the common denominator is \((x-1)^2\).

- Rewrite the Fractions

Rewrite each fraction with the common denominator. The first fraction needs to be multiplied by \( \frac{(x-1)}{(x-1)} \) to match the common denominator: \[ \frac{3x(x-1)}{(x-1)^2} - \frac{x-4}{(x-1)^2} \].

- Simplify the Numerators

Expand and simplify the numerators: \[ \frac{3x(x-1) - (x-4)}{(x-1)^2} = \frac{3x^2 - 3x - x + 4}{(x-1)^2} = \frac{3x^2 - 4x + 4}{(x-1)^2} \].

- Factor the Numerator

Factor the numerator, if possible. However, in this case \(3x^2 - 4x + 4\) does not factor easily, so we leave it as is: \[ \frac{3x^2 - 4x + 4}{(x-1)^2} \].

Key concepts

Factoring Polynomials

Factoring polynomials is a key skill in algebra. It helps break down complex expressions into simpler parts. In this exercise, the polynomial to be factored is the denominator of the second fraction: \(x^2 - 2x + 1\).
Recognize that this can be rewritten. In this case, \(x^2 - 2x + 1\) becomes \((x - 1)^2\).
This insight is crucial. It makes finding common denominators and simplifying easier later on.
To practice, look for other polynomials like \(a^2 - b^2\). Factor this to \((a + b)(a - b)\). Keep practicing by trying different polynomials. Identify patterns and practice factoring as much as possible.
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Common Denominator

Finding a common denominator is necessary for adding and subtracting fractions. Here, the denominators are \(x - 1\) and \((x - 1)^2\).
The common denominator for these two fractions is \((x - 1)^2\).

• For the first fraction, \(\frac{3x}{x-1}\), multiply numerator and denominator by \(x - 1\).
• This transforms it into \(\frac{3x(x - 1)}{(x - 1)^2}\).

This makes it possible to combine the fractions under the common denominator successfully.
Remember, finding the least common denominator simplifies the process of addition or subtraction. Always ensure to multiply both the numerator and the denominator by the same value to maintain the equality of the fraction.
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Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and reducing the expression to its simplest form. In our example, we need to subtract the fractions: \(\frac{3x(x - 1)}{(x - 1)^2} - \frac{x - 4}{(x-1)^2}\).

• First, expand \(3x(x - 1)\) to get \(3x^2 - 3x\).
• Now, combine this with \(- (x - 4)\), taking care to distribute the negative sign properly.

This gives us: \(3x^2 - 3x - x + 4\).
Combine like terms: \(3x^2 - 4x + 4\). This results in \(\frac{3x^2 - 4x + 4}{(x-1)^2}\).
Simplification is crucial. It makes complex expressions easier to manage and solve.
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Algebraic Fractions

Algebraic fractions are fractions with polynomials in the numerator, denominator, or both. They require specific strategies for operations like addition, subtraction, multiplication, and division.
In our example, we deal with subtraction: \(\frac{3x}{x-1} - \frac{x - 4}{(x-1)^2}\). Key steps include:

• Factoring polynomials (if possible) to identify common denominators.
• Rewriting each fraction with a common denominator to prepare for operations.

The final fraction, after simplification, is \(\frac{3x^2 - 4x + 4}{(x-1)^2}\).
Remember, working with algebraic fractions requires patience. Take one step at a time, and practice will make these processes more intuitive.
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