Question

write the partial fraction decomposition of the rational expression.
\\(\frac{x}{x^2 - 4x + 3}\\)

\\(\frac{x}{x^2 - 4x + 3} = \square\\)
(simplify your answer. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Factor the denominator

First, factor the quadratic in the denominator: \(x^2 - 4x + 3=(x - 1)(x - 3)\). So we can write \(\frac{x}{(x - 1)(x - 3)}=\frac{A}{x - 1}+\frac{B}{x - 3}\).

Step2: Multiply both sides by the denominator

Multiply both sides by \((x - 1)(x - 3)\) to get \(x = A(x - 3)+B(x - 1)\).

Step3: Find values of A and B

Let \(x = 1\): Then \(1=A(1 - 3)+B(0)\), so \(1=- 2A\), which gives \(A=-\frac{1}{2}\).
Let \(x = 3\): Then \(3=A(0)+B(3 - 1)\), so \(3 = 2B\), which gives \(B=\frac{3}{2}\).

Step4: Write the partial fraction decomposition

Substitute \(A\) and \(B\) back into \(\frac{A}{x - 1}+\frac{B}{x - 3}\), we get \(\frac{-\frac{1}{2}}{x - 1}+\frac{\frac{3}{2}}{x - 3}=\frac{-1}{2(x - 1)}+\frac{3}{2(x - 3)}\).

Answer:

\(\frac{-1}{2(x - 1)}+\frac{3}{2(x - 3)}\)