The x-intercepts divide the x-axis into intervals, which can be used for solving polynomial inequalities.


The steps for solving a polynomial inequality using graphing are:





-x^3 + 28x + 48 ≥ 0


When there is a negative sign, it is easier to divide the entire function by (-1), however the inequality sign must always be switched at the same time.

x^3 – 28x + 48 ≤ 0

Using the factors of 48, it is found that 2 gives a remainder of 0, therefore it is used for the synthetic division. Note that for synthetic division, there is no “x^2” value, therefore a 0 will be put as placeholder.


(x-2)(x^2 + 2x – 24) ≤ 0


Now this is just simple factoring

(x - 2)(x + 6)(x - 4) ≤ 0


These are the three x-intercepts: X = 2, X = -6 , X = 4


Therefore, the x-intercepts are placed on a number line, and starting from the left to the right, place (+) or (-) signs in the intervals. If the Leading Coefficient is (+) then start with a (+) from the left. If it was negative, then it would start with a (-) sign.


In this case:     (-)      [-6]      (+)      [2]       (-)       [4]       (+)


Now place the x-intercepts on the graph and graph the line according to the signs and intercepts. If one of the intercepts was even (2, 4, 6, 8, etc…) then the sign wouldn't have changed. I.e. if one of the intercepts was ((x-2)^2) then the graph would be (+) or (-) before and after the intercept.

After graphing it is now possible to do what the question was asking, when is the function Less than or Equal to 0. (Note that since we divided by (-1) it is no longer Greater than or Equal to)


The answer would be: f(x) ≤0 when XE(-∞, -6]U[2 , 4]  **The “u” is used o signify “And”. Also for more information about Bracket notation refer here.



If you have any questions, leave them in the comments below.