Expert: Scott A Wilson - 11/5/2009

Question
find the intercepts of the polynominal function f(x). find the intervals for which the graph of f is above the x-axis and below the x-axis.  f(x)=x^3+4x^2-7x-10    the x intercepts are? the y intercepts are?

Answer
The function is x³ + 4x² - 7x - 10.
The first one to try is x=1, which gives 1+4-7-10, which is -12.
The next one to try is x=2.  This gives 8 + 16 - 14 - 10 = 24 - 24 = 0.  Bingo!

The function is (x-2)(x²+6x+5), and that checks out OK.

Now to factor the quadratic, the solutions can be seen to be -1 and -5,
giving (x-2)(x+1)(x+5).

The y intercept is where x = 0, and that is at -10.
The places where y is 0 area at x=2, x=-1, and x=-5.
Since the coefficient on the x³ is positive, this means that to the left it gets really negative and to the right it gets really positive.


From -∞ to -5, the function is below the x axis.
From -5 to -1, the function is above the x axis.
From -1 to 2, the function is below the x axis.
From 2 to ∞, the function is above the x axis.