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

Verified step by step guidance

1. insert step 1> Start by recognizing that the expression (-5 + i)(-5 - i) is a product of two complex conjugates.
2. insert step 2> Use the formula for the product of complex conjugates: (a + bi)(a - bi) = a^2 + b^2. Here, a = -5 and b = 1.
3. insert step 3> Calculate a^2: (-5)^2.
4. insert step 4> Calculate b^2: (1)^2.
5. insert step 5> Add the results from steps 3 and 4 to write the final expression in standard form.

Key Concepts

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 and 'b' is the coefficient of the imaginary unit 'i', which is 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 can use the distributive property (also known as the FOIL method for binomials). This involves multiplying each part of the first complex number by each part of the second, and then combining like terms. The product of two complex conjugates, such as (-5 + i) and (-5 - i), results in a real number.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' is the real part and 'b' is the imaginary part. When multiplying complex numbers, the result should be simplified to this form, which makes it easier to interpret and use in further calculations. In the case of complex conjugates, the imaginary parts cancel out, leading to a purely real result.