Based on the images provided, the student made an error in handling the multiplication of the numerator and denominator when dividing complex numbers. Let me clarify and correct the process.

Problem Breakdown:

1. Expression: $\frac{1+i}{3-i}$
2. To divide, the student should multiply both the numerator and denominator by the conjugate of the denominator ($3+i$).
3. Error Identified:
   • The student correctly used the conjugate $3+i$ but failed to properly expand and simplify the product.
   • The main issue lies in handling $i^2$, as $i^2 = -1$. The student incorrectly added instead of subtracting the imaginary terms' contributions.

Correct Process:

Step 1: Multiply by the Conjugate

$\frac{1+i}{3-i} \cdot \frac{3+i}{3+i} = \frac{(1+i)(3+i)}{(3-i)(3+i)}$

Step 2: Simplify the Denominator

$(3-i)(3+i) = 3^2 - (-1) = 9 + 1 = 10$

Step 3: Expand the Numerator

$(1+i)(3+i) = 1 \cdot 3 + 1 \cdot i + i \cdot 3 + i \cdot i = 3 + i + 3i + i^2$ Since $i^2 = -1$, the numerator becomes: $3 + i + 3i - 1 = 2 + 4i$

Step 4: Write Final Answer

$\frac{2+4i}{10} = \frac{2}{10} + \frac{4i}{10} = \frac{1}{5} + \frac{2i}{5}$

Corrected Answer:

The final simplified answer is: $\boxed{\frac{1}{5} + \frac{2i}{5}}$

Explanation for the Correct Choice:

The correct answer is C:

"When multiplying the numerators and denominators, the student forgot the cross terms that arise from using the Distributive Property and ended up with no imaginary parts."

This is because the student ignored the imaginary contributions from the cross terms and mishandled $i^2$.

Do you want a more detailed explanation or additional practice questions? Here are some related questions to extend your understanding:

1. Why do we multiply by the conjugate when dividing complex numbers?
2. How do you expand and simplify expressions involving $i^2$?
3. What is the importance of simplifying the denominator in complex fraction division?
4. How can errors in handling $i^2$ impact calculations in more complex problems?
5. Can this process be applied to purely imaginary numbers, and how does it simplify?

Tip: Always double-check $i^2$ substitutions to avoid sign errors in your calculations!