Final answer:
To decompose \(\frac{x}{x^2 + x - 6}\) into partial fractions, factor the quadratic to obtain \((x + 3)(x - 2)\) in the denominator. Represent the function as the sum of two simpler fractions, and solve for the constants by setting up equations for specific values of \(x\).
Step-by-step explanation:
The task is to perform a partial fraction decomposition of the function \( \frac{x}{x^2 + x - 6} \). To start, factor the denominator. The quadratic \(x^2 + x - 6\) can be factored into \((x + 3)(x - 2)\). Hence, we can represent the function as:
\[\frac{x}{(x + 3)(x - 2)} = \frac{A}{x + 3} + \frac{B}{x - 2}\]
Where \(A\) and \(B\) are constants to be determined. Multiplying through by the denominator \((x + 3)(x - 2)\), we get:
\[x = A(x - 2) + B(x + 3)\]
By setting up equations for \(x\) at specific values that make one of the terms zero, we can solve for \(A\) and \(B\). Once we find the values of \(A\) and \(B\), the decomposition is complete, and the original function is represented as the sum of simpler fractions.