SOLUTION: The function f is defined as f(x)=(x-2)(x+4)^2. Answer the following questions without using a graphing calculator. a. Use algebraic methods to determine the roots of f. b. O

Question 1126318: The function f is defined as f(x)=(x-2)(x+4)^2. Answer the following questions without using a graphing calculator.
a. Use algebraic methods to determine the roots of f.
b. On what interval(s) of x is f(x)positive? (Hint: enter "U" for ∪ to combine more than one interval.)
c. On what interval(s) of x is f(x)negative? (Hint: enter "U" for ∪ to combine more than one interval.)

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The function value is 0 when any of the factors is 0 -- at x=-4 and at x=2. The function value can change sign only at those two values. So...

(1) On (-infinity,-4), all three factors are negative, so the function value is negative.
(2) On (-4,2) only one factor is negative, so the function value is again negative. (Both factors of (x+4) change sign at x=-4, so the function value remains negative.)
(4) On (2,infinity) all three factors are positive, so the function value is positive. (At x=2, only one factor changes sign, so the function value changes sign.)

So....

ANSWERS:
a. The roots are -4 and 2
b. The function value is positive only on (2,infinity)
c. The function value is negative on (-infinity,-4) U (-4,2)
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