Which of the following is an equation for a polynomial of degree 5 with the following properties:
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 ...
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
Thank you!!!!
To determine which equation corresponds to a polynomial of degree 5 with the given properties, we can analyze each option one by one.
Let's start with option 1: f(x) = -12(x+2)^2(x-3)^3
- Zeros at x=2 and x=-3: The factors (x+2) and (x-3) represent the roots of the polynomial at x= -2 and x=3 respectively. So, this condition is satisfied.
- f(0) = 1296: Plug in x=0 into the equation to determine f(0): f(0) = -12(0+2)^2(0-3)^3 = -12(2)^2(-3)^3 = -12(4)(-27) = 1296. Therefore, this condition is also satisfied.
- f(x) < 0 only on the interval (2, ∞): To check this, let's evaluate f(x) for x values in that interval. Let's try x = 3: f(3) = -12(3+2)^2(3-3)^3 = -12(5)^2(0)^3 = 0. This means f(x) is not negative in the interval (2,∞). So, this condition is not satisfied.
Since the third condition is not satisfied, we can conclude that f(x)= -12(x+2)^2(x-3)^3 is not the correct equation.
Now, let's move on to the second option: f(x) = 12(x-2)^2(x+3)^3
- Zeros at x=2 and x=-3: The factors (x-2) and (x+3) represent the roots of the polynomial at x=2 and x=-3 respectively. So, this condition is satisfied.
- f(0) = 1296: Let's plug in x=0 into the equation to find f(0): f(0) = 12(0-2)^2(0+3)^3 = 12(-2)^2(3)^3 = 12(4)(27) = 1296. Therefore, this condition is satisfied.
- f(x) < 0 only on the interval (2, ∞): Let's evaluate f(x) for x values in that interval. Let's try x = 3: f(3) = 12(3-2)^2(3+3)^3 = 12(1)^2(6)^3 = 12(1)(216) = 2592. This indicates that f(x) is positive in the interval (2,∞). Thus, this condition is not satisfied.
Since the third condition is not satisfied, we can conclude that f(x)= 12(x-2)^2(x+3)^3 is not the correct equation.
Continuing this approach, we can analyze each option in a similar manner.
Considering option 3: f(x) = 27(x-2)^4(x+3)
- Zeros at x=2 and x=-3: This equation satisfies the first condition as it contains factors (x-2) and (x+3) representing the roots at x=2 and x=-3 respectively.
- f(0) = 1296: Let's substitute x=0 into the equation to find f(0): f(0) = 27(0-2)^4(0+3) = 27(-2)^4(3) = 27(16)(3) = 1296. Therefore, this condition is satisfied.
- f(x) < 0 only on the interval (2, ∞): Evaluating f(x) for x values in that interval, let's try x = 3: f(3) = 27(3-2)^4(3+3) = 27(1)^4(6) = 27(1)(6) = 162. Hence, f(x) is positive in the interval (2,∞). This condition is not satisfied.
Since the third condition is not met, we can conclude that f(x)= 27(x-2)^4(x+3) is not the correct equation.
By following this process for each option, we can determine that none of the provided equations satisfy all three conditions. Therefore, the correct equation in the given options is not available.