Question

Find the product of these complex numbers. (6 + 7i)(4 - 2i)

A. 38 + 30i
B. 34 + 22i
C. 40 + 14i
D. 26 - 8i

Answer 1

The product of the complex numbers (6 + 7i) and (4 - 2i) is found by using the distributive property and combining like terms, resulting in 38 + 16i, corresponding to Option B.

Step-by-step explanation:

To find the product of the complex numbers (6 + 7i) and (4 - 2i), we will use the distributive property (also known as the FOIL method in the context of binomials). This involves multiplying each term of the first complex number by each term of the second complex number:

• First, multiply the real parts: 6 × 4 = 24.
• Outer, multiply the real part of the first by the imaginary part of the second: 6 × (-2i) = -12i.
• Inner, multiply the imaginary part of the first by the real part of the second: 7i × 4 = 28i.
• Last, multiply the imaginary parts: 7i × (-2i) = -14i². Since i² = -1, this becomes -14 × (-1) = 14.

Now, combine the like terms (the real parts and the imaginary parts separately):

• Combine the real parts: 24 + 14 = 38.
• Combine the imaginary parts: (-12i + 28i) = 16i.

The final product of the two complex numbers is 38 + 16i, which corresponds to Option B.