In Exercises 9–20, find each product and write the result in standard form. (- 5 + 4i)(3 + i)

검증된 단계별 안내

Recognize that the problem involves multiplying two complex numbers: (-5 + 4i) and (3 + i). Use the distributive property (also known as the FOIL method) to expand the product.

Apply the FOIL method: Multiply the first terms (-5 * 3), the outer terms (-5 * i), the inner terms (4i * 3), and the last terms (4i * i).

Simplify each product: -5 * 3 = -15, -5 * i = -5i, 4i * 3 = 12i, and 4i * i = 4i². Remember that i² = -1, so replace 4i² with 4(-1) = -4.

Combine all the terms: -15 (from the first terms), -5i (from the outer terms), 12i (from the inner terms), and -4 (from the last terms). Group the real parts (-15 and -4) and the imaginary parts (-5i and 12i).

Simplify the expression: Add the real parts (-15 + -4) and the imaginary parts (-5i + 12i) to write the result in standard form a + bi, where a is the real part and b is the coefficient of the imaginary part.

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주요 개념

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Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Multiplication of Complex Numbers

To multiply complex numbers, you apply the distributive property (also known as the FOIL method for binomials) to each part of the numbers. This involves multiplying the real parts and the imaginary parts separately, and then combining like terms, while remembering that i² = -1, which helps simplify the result into standard form.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where a and b are real numbers. In this form, a represents the real part and b represents the imaginary part. When multiplying complex numbers, the final result should be simplified to this standard form for clarity and consistency in mathematical communication.