Final answer:
The correct answer is option c. After performing polynomial division with (x-4), we get a quadratic polynomial, which further factors to (x-2)(x+3). Consequently, the factored form of f(x) is (x-4)(x-2)(x+3), which corresponds to option c).
Step-by-step explanation:
To find the factored form of the given function f(x)=x³−3x²−6x, given that 4 is a zero of the function, we need to use polynomial division or synthetic division to divide the polynomial by (x-4), since x=4 is a zero of the polynomial. Then we can factor the resulting quadratic polynomial to find the other two factors.
First, we divide f(x) by (x-4) and get a quadratic polynomial. We then set this quadratic polynomial equal to zero and factor it, or use the quadratic formula to find the remaining zeros of the polynomial. Once we find the zeros, we can write them as factors in the form of (x-zero). If we get integer zeros, we use those zeros to write the factors.
For this function, after dividing by (x-4), we get the quadratic polynomial x²+1x-6. This polynomial factors to (x-2)(x+3). Therefore, the factored form of the polynomial is f(x)=(x-4)(x-2)(x+3), which corresponds to option c).