Which of the following is an equation for a polynomial of degree 5 with the following properties:

Question

Zeros only at x=2 and x=−3
f(0)=1296
f(x)<0 only on the interval (2,∞)

f(x)=−12(x+2)^2(x−3)^3
f(x)=12(x−2)^2(x+3)^3
f(x)=27(x−2)^4(x+3)
f(x)=−8(x−2)(x+3)^4
f(x)=−27(x+2)^4(x−3)
f(x)=−216(x−2)(x+3)
f(x)=8(x+2)(x−3)^4
f(x)=18(x+2)^3(x−3)^2

check my answer...
Is it
f(x)=27(x−2)^4(x+3)?
thank you ...

Answer 1

no, your equation fails for
f(x)<0 only on the interval (2,∞)
see:
www.wolframalpha.com/input/?i=f%28x%29%3D18%28x%2B2%29%5E3%28x%E2%88%923%29%5E2

start with
f(x) = a(x-2)^2 (x+3)^3, or any of power combinations adding up to 5
f(0) = 1296
a(-2)^2 (3)^3 = 1296
108a = 1296
a = +12

testing: f(x)= 27(x−2)^4(x+3)
f(0) = 27(16)(3) = 1296, so that could work

testing: f(x)=−8(x−2)(x+3)^4
f(0) = -8(-2)(81) = 1296 , ok here as well

testing f(x)=−216(x−2)(x+3), no, exponents don't add to 5

testing f(x) =8(x-2)(x+3)^4 also has f(0) = 1296

so we limited our choices to

f(x)=12(x−2)^2(x+3)^3
f(x)=27(x−2)^4(x+3)
f(x)=−8(x−2)(x+3)^4

looking at x > 2, let's say x = 5
shows that only the last one would be negative

graphing f(x)=−8(x−2)(x+3)^4
will show it to be correct

www.wolframalpha.com/input/?i=f%28x%29%3D-8%28x%E2%88%922%29%28x%2B3%29%5E4

Answer 2

Thank you!!!!