Final answer:
To rewrite the polynomial function q(x) as the product of linear factors, divide q(x) by the given factors (x-1) and (3x-4) using synthetic division. The resulting quotient gives the other two linear factors.
Step-by-step explanation:
To rewrite the polynomial function q(x) as the product of linear factors, we need to divide q(x) by the given factors (x-1) and (3x-4) using synthetic division.
The result of the division will give us the other two factors, which are both linear. Let's perform the synthetic division.
First, divide q(x) by (x-1):
3 14 -8 -49 60
______________________________
(x-1)| 3 1 -7 -56 4
The quotient is 3x^3 + x^2 - 7x - 56.
Next, divide the quotient by (3x-4):
3 1 -7 -56 4
_________________________
(3x-4)| 3 3/4 5/16 -1 0
The resulting quotient is 3x^2 + (3/4)x + (5/16).
Therefore, rewriting q(x) as the product of linear factors, we have q(x) = (x-1)(3x-4)(3x^2 + (3/4)x + (5/16)).