Given a polynomial f(x) and that f(12)=0, apply the Remainder Theorem and then determine which of the following is a factor of f(x)
(x - 3/2)
(x - 0)
(x + 12)
(x - 12)
To apply the Remainder Theorem, we need to divide the polynomial f(x) by each of the given factors and check for a remainder of 0.
1. (x - 3/2): Divide f(x) by (x - 3/2) and check if there is a remainder of 0.
2. (x - 0): Divide f(x) by (x - 0) = x and check if there is a remainder of 0.
3. (x + 12): Divide f(x) by (x + 12) and check if there is a remainder of 0.
4. (x - 12): Divide f(x) by (x - 12) and check if there is a remainder of 0.
Since we are given that f(12) = 0, we can conclude that (x - 12) is a factor of f(x).
To apply the Remainder Theorem, we need to divide the polynomial f(x) by each of the given factors and check if the remainder is zero.
Let's start with the factor (x - 3/2):
1. Divide f(x) by (x - 3/2).
(Use polynomial long division or synthetic division)
Remainder Theorem states that if (x - c) is a factor of f(x), then f(c) = 0.
Therefore, if (x - 3/2) is a factor of f(x), then f(3/2) = 0.
2. Divide f(x) by (x - 0) = x.
Since x is a linear factor, we can directly substitute x = 0 into f(x) to check if it equals zero.
If f(0) = 0, then (x - 0) = x is a factor of f(x).
3. Divide f(x) by (x + 12):
Remainder Theorem states that if (x - c) is a factor of f(x), then f(c) = 0.
Therefore, if (x + 12) is a factor of f(x), then f(-12) = 0.
4. Divide f(x) by (x - 12):
Remainder Theorem states that if (x - c) is a factor of f(x), then f(c) = 0.
Therefore, if (x - 12) is a factor of f(x), then f(12) = 0.
Now, let's determine which of the following factors is a factor of f(x) based on the above calculations:
- Since f(3/2) = 0, (x - 3/2) is a factor of f(x).
- Since f(0) = 0, (x - 0) = x is a factor of f(x).
- Since f(-12) ≠ 0, (x + 12) is not a factor of f(x).
- Since f(12) = 0, (x - 12) is a factor of f(x).
Therefore, the factors of f(x) are (x - 3/2), x, and (x - 12).