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.

Question 1126217: 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 U to combine more than one interval.)
c On what interval(s) of x is f(x) negative? (Hint: enter "U" for U to combine more than one interval.)
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
set f(x) = 0 and you get (x-2) * (x+4)^2 = 0
the roots are x = 2 and x = -4.
f(x) = 0 when x = 2 and x = -4
the intervals between these zero points are:
(-infinity,-4) which means x is smaller than -4.
(-4,2) which means x is greater than -4 and less than 2.
(2,+infinity) which means x is greater than 2.

since the function is continuous, if x is smaller than 0 in an interval, it will stay smaller than x for the whole interval and, if x is greater than 0 in an interval, it will stay greater than 0 for the hole interval.

pick any value in each interval to test whether the function is positive or negative in that interval.

for the interval where x is less than -4, i chose -5.
f(x) = (x-2) * (x+4)^2 becomes f(x) = -7 * 1 which becomes f(x) = -7 which is negative.

for the interval where x is greater than -4 but less than 2, i chose 0.
f(x) = (x-2) * (x+4)^2 becomes f(x) = -2 * 16 which becomes f(x) = -32 which is negative.

for the interval where x is greater than 2, i chose 5.
f(x) = (x-2) * (x+4)^2 becomes f(x) = 3 * 81 which becomes f(x) = 243 which is positive.

the function is negative when x < -4 and when -4 < x < 2.
the function is positive when x > 2.

i interval notation, this results in:

the function is negative during the interval (-infinity, -4) union (-4, 2).
the function is positive during the interval (2, +infinity).

the graph of the function confirms this analysis is true, as shown below.

$$$

RELATED QUESTIONS
The function f is defined as f(x)=(x-2)(x+4)^2. Answer the following questions without... (answered by greenestamps)

The polynomial function f is defined by f(x)x^4-3x^3-2x^2+4x+5. Use a graphing... (answered by stanbon)

The polynomial function f(x) is defined by f(x)=-4x^4+9x^3+2x^2-7x-2. Use a graphing... (answered by lwsshak3)

Use a graphing calculator to approximate the real zeros of the function defined by f(x).... (answered by MathLover1,Alan3354)

Use a calculator to graph f(x). Use the graph to solve f(x) > 0. (Enter your answer using (answered by greenestamps)

How do i simplify equations like f(x)= x^3 - 2x^2 + 6 or f(x)= |(x^2 -4)/(x + 3)| in... (answered by Edwin McCravy)

A function f(x) is defined as f(x) = -8x^2. What is... (answered by richard1234)

f(x)=x^4-3x^3+17x^2+7x-22,[-1,0].Find the turning points using a graphing... (answered by AnlytcPhil)

3 The function f(x) is defined as f(x) = x2 + 2x � 4. The function g(x) is defined as... (answered by josgarithmetic)