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Express the complex fraction below in standard form: \( \frac{1 - 3i}{2 + i} = \_\_ + \_\_ i \).

Express the complex fraction below in standard form: \( \frac{1 - 3i}{2 + i} = \_\_ + \_\_ i \).

Answer

To express the complex fraction \( \frac{1 - 3i}{2 + i} \) in standard form, multiply the numerator and the denominator by the conjugate of the denominator:

  1. Conjugate of \(2 + i\) is \(2 - i\).
  2. Multiply: \( (1 - 3i)(2 - i) = 2 - i - 6i + 3i^2 = 2 - 7i - 3 = -1 - 7i \).
  3. Denominator: \( (2 + i)(2 - i) = 4 - i^2 = 4 + 1 = 5 \).
  4. Result: \( \frac{-1 - 7i}{5} = -\frac{1}{5} - \frac{7}{5}i \).

So, the standard form is \(-\frac{1}{5} + (-\frac{7}{5})i\).

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