Final answer:
The budget polynomial 2a3b - 4a2b + 32ab - 64b can be factored by taking out the common factor of 2b, resulting in the factored form 2b(a3 - 2a2 + 16a - 32).
Step-by-step explanation:
The total budget for Patrick's birthday party is given by the polynomial 2a3b - 4a2b + 32ab - 64b.
To rewrite this polynomial in terms of its factors, you'll want to factor out the greatest common divisor (GCD) from each term. Upon examining the polynomial, we can see that each term has at least one 'b' and some power of 'a'.
Also, each numerical coefficient is divisible by '2'.
So we'll factor out the common terms.
Factoring '2b' from each term, we get:
2b(a3 - 2a2 + 16a - 32)
Next, we check if the remaining polynomial a3 - 2a2 + 16a - 32 can be factored further.
Without any clear common factors or patterns, we might try to factor by grouping or use synthetic division if we know any zeros of the polynomial.
For the purpose of this exercise, we'll assume it cannot be factored further without additional information.
So, the total budget in terms of its factors is 2b(a3 - 2a2 + 16a - 32).