Answer:
x= 1, x= 4, and x= -3
Step-by-step explanation:
Use the possible combinations of factors of the constant term of the polynomial to find a first root. Try 1, -1, 2, -2, 3, -3, etc.
Notice in particular that x = 1 is a root (makes f(1) = 0):
![f(1)=x^3-2*1^2-11*1+12=1-2-11+12=13-13=0](https://tex.z-dn.net/?f=f%281%29%3Dx%5E3-2%2A1%5E2-11%2A1%2B12%3D1-2-11%2B12%3D13-13%3D0)
So we know that x=1 is a root, and therefore, the binomial (x-1) must divide the original polynomial exactly.
As we perform the division, we find that the remainder of it is zero (perfect division) and the quotient is: ![x^2-x-12](https://tex.z-dn.net/?f=x%5E2-x-12)
This is now a quadratic expression for which we can find its factor form:
![x^2-x-12=x^2-4x+3x-12=x(x-4)+3(x-4)=(x-4)*(x+3)](https://tex.z-dn.net/?f=x%5E2-x-12%3Dx%5E2-4x%2B3x-12%3Dx%28x-4%29%2B3%28x-4%29%3D%28x-4%29%2A%28x%2B3%29)
From the factors we just found, we conclude that x intercepts (zeroes) of the original polynomial are those x-values for which each of the factors: (x-1), (x-4) and (x+3) give zero. That is, the values x= 1, x= 4, and x= -3. (these are the roots of the polynomial.
Mark these values on the number line as requested.