Find each product. Write answers in standard form. (3-i)(3+1)(2-6i)

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First, identify the expression to be simplified: \((3 - i)(3 + 1)(2 - 6i)\).

Notice that \((3 + 1)\) is a real number and can be simplified directly to \(4\). So rewrite the expression as \((3 - i) \times 4 \times (2 - 6i)\).

Next, multiply the complex numbers \((3 - i)\) and \((2 - 6i)\) using the distributive property (FOIL method): \((3 - i)(2 - 6i) = 3 \times 2 + 3 \times (-6i) - i \times 2 - i \times (-6i)\).

Simplify each term: \(3 \times 2 = 6\), \(3 \times (-6i) = -18i\), \(-i \times 2 = -2i\), \(-i \times (-6i) = +6i^2\). Remember that \(i^2 = -1\), so \(6i^2 = 6 \times (-1) = -6\).

Combine like terms from the multiplication: Real parts: \(6 - 6 = 0\), Imaginary parts: \(-18i - 2i = -20i\). So, \((3 - i)(2 - 6i) = -20i\). Finally, multiply this result by \(4\) to get the product in standard form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers and Imaginary Unit

Complex numbers have a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to work with i is essential for simplifying expressions involving complex numbers.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL method) and combining like terms, remembering to replace i² with -1. This process allows the product to be expressed in the standard form a + bi.

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a and b are real numbers. Writing answers in this form means separating the real and imaginary parts clearly after performing operations.