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A proper fraction is a type of fractional expression where the degree of the numerator is less than the degree of the denominator. In mathematical terms, the degree is the highest power of the variable present in the polynomial. For example, in the expression \(\frac{-1}{x^2-3x}\), the numerator, \(-1\), is a constant and therefore of degree 0, while the denominator \(x^2-3x\) is of degree 2. Consequently, since 0 is less than 2, this fraction is a proper fraction. Understanding the nature of proper fractions is essential because they qualify for partial fraction decomposition, which is a process used to break down complex rational expressions into simpler fractions, making them easier to integrate or simplify further.

Factoring is a fundamental technique used to simplify expressions, particularly useful in the context of partial fraction decomposition. For the expression \(\frac{-1}{x^2-3x}\), the denominator \(x^2-3x\) needs to be factored to facilitate decomposition. Here are some quick tips to factor polynomials effectively:

• Look for common factors: In \(x^2-3x\), notice the common factor \(x\).
• Apply distributive law "in reverse": This helps split terms smoothly. After pulling \(x\) out, the expression becomes \(x(x - 3)\).
• Confirm by distributing back: Multiplying \(x(x - 3)\) again should yield \(x^2-3x\).

Using these techniques, you can break down complex expressions into simpler parts suitable for partial fractions. This offers clarity and simplifies subsequent mathematical operations.

Linear factors are components of a polynomial that are first-degree, meaning they are of the form \(ax + b\), where \(a\) and \(b\) are constants and \(a eq 0\). In partial fraction decomposition, identifying linear factors in the denominator is crucial as it dictates the structure of the decomposition.For the fraction \(\frac{-1}{x^2-3x}\), once factored, the denominator becomes \(x(x - 3)\). These are both linear factors:

• \(x\) is a linear factor because it is effectively \(1x + 0\).
• \(x - 3\) is also a linear factor of the form \(1x - 3\).

Recognizing these linear factors allows us to determine the form of the partial fraction decomposition as \(\frac{A}{x} + \frac{B}{x - 3}\). Each factor corresponds to a term in the decomposition, with constants \(A\) and \(B\) waiting to be solved for.