Given a polynomial f(x) and that f(12)=0, apply the remainder theorem and then determine which of the fallowing is a factor of f(x)
(x-12)
(x-3/2)
(x+12)
(x-0)
According to the Remainder Theorem, if f(12) = 0, then (x - 12) is a factor of f(x).
However, we cannot determine whether (x - 3/2), (x + 12), or (x - 0) are factors of f(x) based on the information given in the question.
To apply the remainder theorem, we can divide the polynomial f(x) by (x-12). If the remainder is equal to zero, then (x-12) is a factor of f(x).
Let's proceed with the division:
Step 1: Write the polynomial f(x) and the divisor (x-12):
f(x) = ... (given polynomial)
Divisor: (x-12)
Step 2: Perform the division:
Long division:
```
... (quotient)
______________________________________
x - 12 | f(x) = ...
- (f(x) terms)
______________________________________
(remainder)
```
Step 3: Since f(12) = 0 and the remainder is zero, we can conclude that (x-12) is a factor of f(x).
Now, let's determine which of the given options is a factor of f(x):
1. (x-12): This is already confirmed to be a factor.
2. (x-3/2): Since (x-3/2) was not given as a possible factor, we cannot confirm if it is a factor or not based on the information provided.
3. (x+12): We only know that (x-12) is a factor, not (x+12). Therefore, we cannot determine if (x+12) is a factor or not.
4. (x-0): (x-0) simplifies to x, which is always a factor of any polynomial. Therefore, (x-0) is a factor of f(x).
In conclusion, the factors of f(x) are:
- (x-12)
- (x-0) or simply x.
To use the remainder theorem, we divide the polynomial by each potential factor and check if the remainder is zero. Let's go through the process for each of the given factors.
1. (x-12):
To check if (x-12) is a factor of f(x), we need to divide f(x) by (x-12) and see if the remainder is zero.
Step 1: Divide f(x) by (x-12) using polynomial long division or synthetic division.
For example, let's say f(x) = 2x^3 - 3x^2 + 5x - 6.
2x^2 + 21x + 246
_____________________
(x - 12) | 2x^3 - 3x^2 + 5x - 6
- (2x^3 - 24x^2)
___________________
21x^2 + 5x
- (21x^2 - 252x)
___________________
5x - 6
- (5x - 60)
___________________
54
Step 2: Check the remainder. In this case, the remainder is 54, which is non-zero. Therefore, (x-12) is not a factor of f(x).
2. (x-3/2):
Again, we need to divide f(x) by (x-3/2) and check the remainder.
Step 1: Divide f(x) using polynomial long division or synthetic division.
Step 2: Check the remainder. Since the division process is not provided, we cannot determine whether the remainder is zero or not.
3. (x+12):
Once again, we divide f(x) by (x+12).
Step 1: Divide f(x) by (x+12) using polynomial long division or synthetic division.
2x^2 - 27x + 234
_______________________
(x + 12) | 2x^3 - 3x^2 + 5x - 6
- (2x^3 + 24x^2)
_____________________
- 27x^2 + 5x
- (-27x^2 - 324x)
____________________
29x - 6
- (29x + 348)
______________________
- 354
Step 2: Check the remainder. In this case, the remainder is -354, which is non-zero. Therefore, (x+12) is not a factor of f(x).
4. (x-0):
Since (x-0) simplifies to x, we can just evaluate f(x) at x = 0.
f(x) = 2x^3 - 3x^2 + 5x - 6
f(0) = 2(0)^3 - 3(0)^2 + 5(0) - 6
= 0 - 0 + 0 - 6
= -6
Since f(0) is not equal to zero, (x-0) is not a factor of f(x).
In conclusion, none of the given factors (x-12), (x-3/2), (x+12), or (x-0) are factors of f(x) because none of them yield a remainder of zero when f(x) is divided by them.