Question

23 which function has a zero with a multiplicity of 2?
a $f(x) = (x + 2)(x-2)$
b $f(x) = (x-2)(x-2)(x-2)$
c $f(x) = (x-1)(x + 3)$
d $f(x) = (x-5)(x + 1)(x-5)$

Explanation:

Step1: Define zero multiplicity

The multiplicity of a zero is the number of times its corresponding factor repeats in the function's factored form.

Step2: Analyze Option A

$f(x)=(x+2)(x-2)$ has zeros $x=-2$ (multiplicity 1) and $x=2$ (multiplicity 1). No zero with multiplicity 2.

Step3: Analyze Option B

$f(x)=(x-2)(x-2)(x-2)$ has zero $x=2$ with multiplicity 3. Not 2.

Step4: Analyze Option C

$f(x)=(x-1)(x+3)$ has zeros $x=1$ (multiplicity 1) and $x=-3$ (multiplicity 1). No zero with multiplicity 2.

Step5: Analyze Option D

$f(x)=(x-5)(x+1)(x-5)$ can be rewritten as $f(x)=(x-5)^2(x+1)$. It has zero $x=5$ with multiplicity 2, and $x=-1$ with multiplicity 1.

Answer:

D. $f(x) = (x-5)(x + 1)(x-5)$