Answer:
To show that a x - 4 is a factor of the function f(x) = 2x^3 - 5x^2 - 11x - 4, we need to show that f(a x - 4) = 0 for all values of x.
We can use polynomial long division or synthetic division to divide f(x) by (a x - 4) and check if the remainder is zero. Here, we will use synthetic division:
4/a | 2 -5 -11 -4
| 8a 12a 4a-16
+------------------
2 8a-5 a-11 4a-20
The last term in the bottom row of the table is the remainder, which is 4a - 20. We can see that this remainder is equal to zero if a = 5.
Therefore, if we substitute x = (5/ax - 4) into the function f(x), we get:
f(a x - 4) = 2(5/ax - 4)^3 - 5(5/ax - 4)^2 - 11(5/ax - 4) - 4
Simplifying this expression, we get:
f(a x - 4) = 0
This means that a x - 4 is indeed a factor of the function f(x) = 2x^3 - 5x^2 - 11x - 4 when a = 5.