Problem 11 - College Algebra

Question

Find each product and write the result in standard form. $$ (-5+4 i)(3+i) $$

Step-by-Step Solution

Answer

The product of \(-5+4i\) and \(3+i\) in standard form is \(-19 + 7i\).

Step 1: Multiplication using FOIL method

Just like expanding expressions with real numbers, proceed with the standard FOIL method. Multiply the First terms, Outside terms, Inside terms, and Last terms. \[(-5+4i) (3+i) = (-5)(3) + (-5)(i) + (4i)(3) + (4i)(i)\]

Step 2: Simplify the multiplied equation

Compute the products and simplify the equation, bearing in mind that \(i^2 = -1\). \[-15 - 5i + 12i - 4i^2 = -15 + 7i + 4(-1)\]

Step 3: Replace \(i^2\) with \(-1\) and simplify to standard form

Replace \(i^2\) with \(-1\) and simplify the equation to result in the standard form. \[-15 + 7i - 4 = -19 + 7i\]

Key Concepts

FOIL Method

The FOIL method is an acronym that stands for First, Outside, Inside, Last. This technique is used to multiply two binomials and is particularly handy when dealing with complex numbers. When multiplying complex numbers such as \( (-5+4i)(3+i) \), each part of the first complex number is multiplied by each part of the second one. Let's break it down:

• First: Multiply the first terms from each binomial (\(-5 * 3\)).
• Outside: Multiply the outer terms (\(-5 * i\)).
• Inside: Multiply the inner terms (\(4i * 3\)).
• Last: Multiply the last terms from each binomial (\(4i * i\)).

After applying the FOIL method, we combine the like terms, ensuring to replace the square of the imaginary unit \(i^2\) with \(-1\), to keep the solution in a simplified form.

Standard Form of Complex Numbers

Complex numbers are expressed in the format \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This form, called the standard form, helps to easily identify the real and imaginary parts of the complex number. For example, the product \(-19 + 7i\) is in standard form with the real part being \(-19\) and the imaginary part being \(7i\). When multiplying complex numbers, the goal is to simplify the result into this standard form, combining like terms and simplifying expressions involving \(i\), particularly remembering to use the fact that \(i^2 = -1\). This keeps the calculations tidy and the numbers ready for further algebraic manipulation or graphical representation on the complex plane.

Imaginary Units

The imaginary unit, represented by \(i\), is a mathematical concept used to extend the real number system to include solutions to equations that cannot be solved using real numbers alone. Specifically, \(i\) is defined by the property that \(i^2 = -1\). This unique characteristic allows for the expression of numbers as a combination of a real and an imaginary part. When multiplying complex numbers, it is crucial to recognize when \(i^2\) appears, as it must be replaced with \(-1\) to maintain the calculations within the realm of complex numbers and adhere to the standard form. For instance, in the product \(4i * i\), we find \(i^2\) which simplifies to \(-1\), essential for reaching a proper conclusion with complex numbers.