Answer:
<h2>x = 0 and x = 2</h2>
Step-by-step explanation:
![\text{The domain:}\\\\(x+3)(x-5)\neq0\iff x+3\neq0\ \wedge\ x-5\neq0\\\\x\neq-3\ \wedge\ x\neq5\\\\========================\\\\f(x)=\dfrac{x(x-2)}{(x+3)(x-5)}\\\\\text{The zeros are for}\ f(x)=0\\\\\dfrac{x(x-2)}{(x+3)(x-5)}=0\iff x(x-2)=0\iff x=0\ \vee\ x-2=0\\\\x=0\in D\ \vee\ x=2\in D](https://tex.z-dn.net/?f=%5Ctext%7BThe%20domain%3A%7D%5C%5C%5C%5C%28x%2B3%29%28x-5%29%5Cneq0%5Ciff%20x%2B3%5Cneq0%5C%20%5Cwedge%5C%20x-5%5Cneq0%5C%5C%5C%5Cx%5Cneq-3%5C%20%5Cwedge%5C%20x%5Cneq5%5C%5C%5C%5C%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%5C%5C%5C%5Cf%28x%29%3D%5Cdfrac%7Bx%28x-2%29%7D%7B%28x%2B3%29%28x-5%29%7D%5C%5C%5C%5C%5Ctext%7BThe%20zeros%20are%20for%7D%5C%20f%28x%29%3D0%5C%5C%5C%5C%5Cdfrac%7Bx%28x-2%29%7D%7B%28x%2B3%29%28x-5%29%7D%3D0%5Ciff%20x%28x-2%29%3D0%5Ciff%20x%3D0%5C%20%5Cvee%5C%20x-2%3D0%5C%5C%5C%5Cx%3D0%5Cin%20D%5C%20%5Cvee%5C%20x%3D2%5Cin%20D)
Theorem: If a function y = f(x) has a real root of b, then (x – b) is a factor of f(x).
As given in the problem, there are two roots: –2 and 1/2. The multiplicity of 1/2 is 2 meaning that the root 1/2 repeats twice. So the function f(x) can be written like this.
f(x) = k• (x – (–2))(x – 1/2)^2 = k•(x + 2)(x – 1/2)^2
We're supposed to find the coefficient k to complete the function.
Given that f(–3) = 5, we can plug –3 in for x and 5 in for f(x).
So 5 = k •(–3 + 2)(–3 – 1/2)^2
5 = k(–1)(–7/2)^2
5 = -k•49/4
Then 5 • 4/49 = -k
Or k = –20/49
So the function with the least degree is
f(x) = –20/49 (x + 2)(x – 1/2)^2.
Answer:
3x^2-7x-20
Step-by-step explanation:
Answer:
It will be the 26th divided by 7