Use the remainder theorem to determine if the given number $$ c $$ is a zero of the polynomial. $$ q(x)=x^{3}-4 x^{2}+4 x-16 $$ (a) $$ c=2 i $$ (b) $$ c=-2 i $$

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1. The remainder theorem states that if we substitute $$ c $$ into the polynomial $$ q(x) $$ and obtain 0, then $$ c $$ is a zero of $$ q(x) $$.

2. Given the polynomial  
   $$
   q(x)=x^{3}-4 x^{2}+4 x-16,
   $$  
   we will evaluate $$ q(c) $$ for each value of $$ c $$.

3. (a) For $$ c=2i $$:

- Compute $$(2i)^3$$:  
     $$
     (2i)^3 = 8i^3.
     $$  
     Since $$ i^3 = -i $$,  
     $$
     8i^3 = 8(-i) = -8i.
     $$
     
   - Compute $$-4(2i)^2$$:  
     $$
     (2i)^2 = 4i^2.
     $$  
     Since $$ i^2 = -1 $$,  
     $$
     4i^2 = 4(-1) = -4.
     $$  
     Thus,  
     $$
     -4(2i)^2 = -4(-4) = 16.
     $$
     
   - Compute $$4(2i)$$:  
     $$
     4(2i)=8i.
     $$
     
   - Constant term: $$-16$$.

- Combine all terms:  
     $$
     q(2i)=(-8i)+(16)+(8i)+(-16).
     $$
     
   - Simplify by combining like terms:  
     $$
     (-8i+8i)+(16-16)=0+0=0.
     $$
     
   Since $$ q(2i)=0 $$, by the remainder theorem, $$ 2i $$ is a zero of $$ q(x) $$.

4. (b) For $$ c=-2i $$:

- Compute $$(-2i)^3$$:  
     $$
     (-2i)^3 = -8i^3.
     $$  
     Since $$ i^3 = -i $$,  
     $$
     -8i^3 = -8(-i) = 8i.
     $$

- Compute $$-4(-2i)^2$$:  
     $$
     (-2i)^2 = 4i^2.
     $$  
     Since $$ i^2 = -1 $$,  
     $$
     4i^2 = 4(-1) = -4.
     $$  
     Thus,  
     $$
     -4(-2i)^2 = -4(-4)=16.
     $$
     
   - Compute $$4(-2i)$$:  
     $$
     4(-2i)=-8i.
     $$
     
   - Constant term: $$-16$$.

- Combine all terms:  
     $$
     q(-2i)=(8i)+(16)+(-8i)+(-16).
     $$
     
   - Simplify by combining like terms:  
     $$
     (8i-8i)+(16-16)=0+0=0.
     $$  
     
   Since $$ q(-2i)=0 $$, by the remainder theorem, $$ -2i $$ is also a zero of $$ q(x) $$.

5. Final Answer:

Both $$ c=2i $$ and $$ c=-2i $$ are zeros of the polynomial $$ q(x)=x^{3}-4x^{2}+4x-16 $$.
Both $$ c=2i $$ and $$ c=-2i $$ are zeros of the polynomial $$ q(x)=x^{3}-4x^{2}+4x-16 $$.

Use the remainder theorem to determine if the given number \( c \) is a zero of the polynomial. \( q(x)=x^{3}-4 x^{2}+4 x-16 \) (a) \( c=2 i \) (b) \( c=-2 i \)

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