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) .(1 point)
1. (x-12)
2. (x+12)
3. (x-0)
4. (x- 3/2)
The Remainder Theorem states that if a polynomial f(x) is divided by x - c, then the remainder is equal to f(c).
Since f(12) = 0, this means that the remainder when f(x) is divided by x - 12 is 0. Therefore, (x-12) is a factor of f(x).
Out of the given options, the factor that matches this is 1. (x-12).
To apply the Remainder Theorem, we need to find the remainder when the polynomial f(x) is divided by the given factor.
In this case, we know that f(12) = 0. According to the Remainder Theorem, if the factor (x-a) is a factor of f(x), then f(a) = 0.
Let's check the options:
1. (x - 12): f(12) = 12 - 12 = 0
2. (x + 12): f(-12) ≠ 0, so it is not a factor.
3. (x - 0): f(0) ≠ 0, so it is not a factor.
4. (x - 3/2): f(3/2) ≠ 0, so it is not a factor.
Therefore, the factor of f(x) is (x - 12), which means option 1 is the correct answer.
To apply the Remainder Theorem and determine which of the given options is a factor of the polynomial f(x), we need to use the fact that f(12) = 0. The Remainder Theorem states that if a polynomial f(x) is divided by (x - k), then the remainder will be equal to f(k).
In this case, since we know that f(12) equals zero, we can conclude that the remainder when f(x) is divided by (x - 12) is also zero. Hence, option 1, (x - 12), is a factor of f(x).
To verify this, you can divide f(x) by (x - 12) using polynomial long division or synthetic division. If the remainder is zero, then (x - 12) is indeed a factor. If the remainder is non-zero, then (x - 12) is not a factor.
Therefore, the correct answer is option 1, (x - 12).