Bot GPT 4-o mini To multiply the complex numbers \((2−5i)(3+i)\), we can use the distributive property (also known as the FOIL method for binomials).
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First: Multiply the first terms: \[ 2 \cdot 3 = 6 \]
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Outer: Multiply the outer terms: \[ 2 \cdot i = 2i \]
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Inner: Multiply the inner terms: \[ -5i \cdot 3 = -15i \]
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Last: Multiply the last terms: \[ -5i \cdot i = -5i^2 \]
Since \(i^2 = -1\), we have: \[ -5i^2 = -5(-1) = 5 \]
Now, combine all these results together: \[ 6 + 2i - 15i + 5 \] Combine the real parts and the imaginary parts: \[ (6 + 5) + (2i - 15i) = 11 - 13i \]
So, the final result is: \[ \boxed{11 - 13i} \]
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