Final answer:
The original rational function 1 / [(x² - x - 6)(x² + 6x + 9)] can be rewritten as 1 / [(x - 3)(x + 2)(x + 3)²] by factoring the quadratics in the denominator.
Step-by-step explanation:
The student has asked to rewrite a rational function as a product of powers of linear terms and irreducible quadratic terms. The function to be factored is 1 / [(x² - x - 6)(x² + 6x + 9)]. First, we need to factor the quadratic expressions in the denominator.
The first quadratic expression x² - x - 6 can be factored into (x - 3)(x + 2), because -3 and 2 are the roots of the quadratic equation found by setting x² - x - 6 = 0. The second quadratic expression x² + 6x + 9 is already a perfect square, which can be expressed as (x + 3)².
Putting it all together, our original rational function becomes:
1 / [(x - 3)(x + 2)(x + 3)²].
This expression is already in the form of a product of powers of linear terms and irreducible quadratic terms, as the first two factors are linear and the last factor is irreducible since it's a perfect square.