Steve S. answered • 01/31/14

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Tutoring in Precalculus, Trig, and Differential Calculus

f(x)=-8((x+6)^5)((x+3)^9)((x-1)^7)

The terms "root" and "zero" are often used interchangeably but it is more correct to use "zero" when talking about functions. A "zero" is an x-value, say x = c, for which f(c) = 0.

In this case, f(c) = 0 when c = -6, -3, or 1 (using the Zero Product Property).

The end behavior to the right follows the sign of the leading coefficient. For f(x) that's -8, so its end behavior to the right is down.

The degree of f(x) is 5+9+7 = 21 which is odd. Odd degree polynomials have opposite end behavior; e.g., a straight line has degree 1 which is odd.

So for f(x) the end behavior to the left is up.

The multiplicity of each factor of f(x) is odd, so the graph will go through the x-axis at that zero.

Putting this all together:

above , -6 , below , -3 , above , 1 , below