Answer:
The values of the functions are a) f(-3) = 12, b) f(-6) = 0, d) f(3) = 25, Option (a) is true, option (b) is true, and option (d) is true.
Explanation:
To evaluate each function at the given value using synthetic substitution, we can perform the synthetic substitution process for each function.
Function: f(a) = a^3 + 5a^2 at a = -3
Substitute -3 into the function:
Set up the synthetic division table: -3 | 1 5 0
Perform the synthetic division: -3 | 1 5 0 |___ 1 2 0
The result is the constant at the bottom of the synthetic division, which is 0.
So, the correct answer is: f(-3) = 0
Function: f(x) = x^6 + 2x^5 - 26x^4 - 12x^3 - 3x^2 - 14x + 12 at x = -6
Substitute -6 into the function:
Set up the synthetic division table: -6 | 1 2 -26 -12 -3 -14 12
Perform the synthetic division: -6 | 1 2 -26 -12 -3 -14 12 |___ 1 -4 8 0 3 -2 0
The result is the constant at the bottom of the synthetic division, which is 0.
So, the correct answer is: f(-6) = 0
Function: f(a) = a^6 - 9a^5 + 22a^4 - 10a^3 - 3a^2 - 15a + 25 at a = 3
Substitute 3 into the function:
Set up the synthetic division table: 3 | 1 -9 22 -10 -3 -15 25
Perform the synthetic division: 3 | 1 -9 22 -10 -3 -15 25 |___ 1 -6 4 2 3 0 25
The result is the constant at the bottom of the synthetic division, which is 25.
So, the correct answer is: f(3) = 25
Thus, Option (a) is true, option (b) is true, option (d) is true.