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What is the value of left-parenthesis 7 minus 3 i right-parenthesis left-parenthesis 6 plus i right-parenthesis?

Answer options with 4 options
A.
42 minus 14 i

B.
45 minus 11 i

C.
45 minus 25 i

D.
42 minus 28 i

1 answer

To find the value of \((7 - 3i)(6 + i)\), we can use the distributive property (also known as the FOIL method for binomials).

Let's calculate it step by step:

  1. First: Multiply the first terms: \[ 7 \cdot 6 = 42 \]

  2. Outer: Multiply the outer terms: \[ 7 \cdot i = 7i \]

  3. Inner: Multiply the inner terms: \[ -3i \cdot 6 = -18i \]

  4. Last: Multiply the last terms: \[ -3i \cdot i = -3i^2 \] Since \(i^2 = -1\), we have: \[ -3i^2 = -3(-1) = 3 \]

Now, we can combine all these results: \[ (7 - 3i)(6 + i) = 42 + 7i - 18i + 3 \]

Combining like terms:

  • Real part: \(42 + 3 = 45\)
  • Imaginary part: \(7i - 18i = -11i\)

So, we have: \[ (7 - 3i)(6 + i) = 45 - 11i \]

Thus, the answer is B. \(45 - 11i\).