Factor completely. then sketch the graph with at least six points, including all roots and the y-intercept.

The rational root theorem suggests the following possible rational roots:

22, -22, 11, -11, 2, -2, 1, -1

(obtained by dividing all divisors of the constant term, 22, by the coefficient of the leading term, 1). Checking all of these possibilities, we happen to find that .

By the polynomial remainder theorem (if for some polynomial , then divides ), we then know that and are factors.

Performing the division gives

The remaining quadratic has discriminant

which means the remaining two terms are complex and the quadratic cannot be factored any further (over the real numbers). So

set each factor equal to 0

4x-3 = 0

2x-1 = 0

solve for x.

4x-3=0

4x=3

x= 3/4

solve for x. 

2x-1= 0

2x= 1

x= 1/2

eventually after a few more steps, your answer should be 1/2 is less than or equal to x and 3/4 is greater than or equal to x