In Exercises 9–20, find each product and write the result in standard form. (−5 + i)(−5 − i)

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Recognize that the expression \((−5 + i)(−5 − i)\) is a product of two complex conjugates. Complex conjugates have the form \(a + bi\) and \(a - bi\).

Recall the formula for the product of complex conjugates: \((a + bi)(a - bi) = a^2 + b^2\). This results in a real number because the imaginary parts cancel out.

Identify \(a = -5\) and \(b = 1\) from the given expression \((−5 + i)(−5 − i)\).

Apply the formula by squaring \(a\) and \(b\) and then adding them: calculate \((-5)^2 + (1)^2\).

Write the result in standard form, which for complex numbers is \(x + yi\), but since the product of conjugates is real, the imaginary part will be zero.

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주요 개념

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Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. The standard form refers to writing the result explicitly as a sum of a real number and an imaginary number. Understanding this form is essential for interpreting and simplifying complex number expressions.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL method) and applying the rule i² = -1. Each term in the first complex number is multiplied by each term in the second, and like terms are combined to simplify the expression.

Imaginary Unit Properties

The imaginary unit i is defined such that i² = -1. This property is crucial when simplifying products involving i, as it allows conversion of i² terms into real numbers, enabling the expression to be written in standard form.