Expand the quotients in Exercises 1–8 by partial fractions.
z / (z³ - z² - 6z)

Verified step by step guidance

1. Start by factoring the denominator of the given expression \(\frac{z}{z^{3} - z^{2} - 6z}\). First, factor out the common factor \(z\): \(z^{3} - z^{2} - 6z = z(z^{2} - z - 6)\).
2. Next, factor the quadratic expression \(z^{2} - z - 6\). Find two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\), so: \(z^{2} - z - 6 = (z - 3)(z + 2)\).
3. Rewrite the original expression using the factored denominator: \(\frac{z}{z(z - 3)(z + 2)}\).
4. Set up the partial fraction decomposition. Since the denominator factors are linear and distinct, express the fraction as: \(\frac{z}{z(z - 3)(z + 2)} = \frac{A}{z} + \frac{B}{z - 3} + \frac{C}{z + 2}\), where \(A\), \(B\), and \(C\) are constants to be determined.
5. Multiply both sides of the equation by the common denominator \(z(z - 3)(z + 2)\) to clear the denominators: \(z = A(z - 3)(z + 2) + B z (z + 2) + C z (z - 3)\). This equation can now be expanded and simplified to solve for \(A\), \(B\), and \(C\) by equating coefficients or substituting convenient values of \(z\).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions, making integration or other operations easier. It involves factoring the denominator and writing the original fraction as a sum of fractions with those factors as denominators.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into products of simpler polynomials. For partial fractions, factoring the denominator completely into linear or irreducible quadratic factors is essential to set up the decomposition correctly.

Solving Systems of Equations

After setting up the partial fractions, coefficients are found by equating numerators and solving the resulting system of linear equations. This step is crucial to determine the unknown constants in the decomposition.