Multiply (3+2i)(1-i) and express in a+bi form; then convert to polar form r(cos φ + i sin φ).

• A. Product = 5 - i; r = √26; φ = -arctan(1/5) ≈ -0.1974 rad
• B. Product = 5 - i; r = √26; φ = arctan(5)
• C. Product = 5 + i; r = √26; φ = -arctan(1/5)
• D. Product = -5 + i; r = √26; φ = π - arctan(1/5)

Answer: (A)

Multiplying complex numbers uses distributivity and the fact that i^2 = −1. Compute (3+2i)(1−i) = 3(1−i) + 2i(1−i) = 3 − 3i + 2i − 2i^2 = 3 − i + 2 = 5 − i. So the product in a+bi form is 5 − i.

To convert to polar form r(cos φ + i sin φ), first find the magnitude: r = sqrt(a^2 + b^2) = sqrt(5^2 + (−1)^2) = sqrt(26). The angle φ is the argument, given by φ = arctan(b/a) with attention to the quadrant. Here a = 5 and b = −1, so φ = arctan(−1/5) = −arctan(1/5). The point (5, −1) lies in the fourth quadrant, so the principal angle is a small negative value, approximately −0.1974 radians.

Thus the polar form is sqrt(26) [cos(−arctan(1/5)) + i sin(−arctan(1/5))], which matches the description: product 5 − i, r = √26, φ = −arctan(1/5) ≈ −0.1974 rad.