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What is the product of left-parenthesis negative 7 plus 4 i right-parenthesis times left-parenthesis 1 minus 8 i right-parenthesis?

Answer options with 4 options
A.
negative 39 minus 52 i

B.
negative 39 plus 60 i

C.
25 minus 52 i

D.
25 plus 60 i

1 answer

To find the product of \(( -7 + 4i )\) and \(( 1 - 8i )\), we can use the distributive property (also known as the FOIL method for binomials).

\[ (-7 + 4i)(1 - 8i) \]

Calculating this step by step:

  1. First Term: \[ -7 \cdot 1 = -7 \]

  2. Outer Term: \[ -7 \cdot (-8i) = 56i \]

  3. Inner Term: \[ 4i \cdot 1 = 4i \]

  4. Last Term: \[ 4i \cdot (-8i) = -32i^2 \]

    Since \(i^2 = -1\), we have: \[ -32i^2 = -32(-1) = 32 \]

Now combine all the results: \[ -7 + 56i + 4i + 32 \]

Combining the real parts and the imaginary parts:

  • Real parts: \(-7 + 32 = 25\)
  • Imaginary parts: \(56i + 4i = 60i\)

Putting it all together: \[ 25 + 60i \]

Thus, the product is: \[ \boxed{25 + 60i} \]

The correct answer is D. 25 plus 60 i.