Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (-1+2 i)(-2+3 i) $$

Step-by-Step Solution

Answer

-4 - 7i

Step 1: Apply the Distributive Property

To multiply the two complex numbers (-1 + 2i) and (-2 + 3i), use the distributive property (also known as the FOIL method for binomials). First, multiply each term in the first complex number by each term in the second complex number. Calculate each product individually: 1. (-1) imes (-2) = 2 2. (-1) imes (3i) = -3i 3. (2i) imes (-2) = -4i 4. (2i) imes (3i) = 6i^2 Combine these results for the intermediate expression: 2 - 3i - 4i + 6i^2.

Step 2: Simplify the Intermediate Expression

Now, simplify the expression from Step 1: 2 - 3i - 4i + 6i^2. Begin by combining like terms: the real numbers and the imaginary numbers. The imaginary numbers are -3i and -4i: -3i - 4i = -7i. Now plug these back into the expression: 2 - 7i + 6i^2. Next, simplify 6i^2. Recall that i^2 = -1, so: 6(-1) = -6. Replace 6i^2 with -6 in the expression: 2 - 7i - 6.

Step 3: Combine Real Numbers

Finally, combine the real parts of the expression: 2 - 6. This results in -4. The expression is now fully simplified: -4 - 7i.

Key Concepts

Distributive Property

Complex numbers, like any other algebraic expressions, can utilize the distributive property for multiplication. The distributive property states that for any numbers or expressions, you can spread the multiplication over addition. For example, in simple terms,

• if you have (a + b)(c + d), it expands to a*c + a*d + b*c + b*d.

When dealing with complex numbers, the property becomes crucial since each number has a real and an imaginary part. Consider (-1 + 2i)(-2 + 3i): each term in the first expression is multiplied by each term in the second expression. This step might resemble multiplying two binomials. Hence, each of these products is found individually and then added together for the complete expanded form.

Imaginary Unit

The imaginary unit, denoted as i, is a crucial concept in complex numbers. It is defined as the square root of -1. Because of this definition, the square of i, which is i^2, equals -1.

• This extension of number systems allows representation of numbers beyond the real number line.
• A complex number can be expressed as a + bi, where a is the real part, and bi is the imaginary part.

In our problem, i inherently changes how multiplication and simplification work. For example, when we encounter 6i^2 in the solution, we use the definition of i^2 to transform it into a real number. Realizing this relation is essential to reducing mistakes and further simplification.

FOIL Method

The FOIL method is a systematic way to multiply two binomials. FOIL stands for First, Outer, Inner, Last, crucial terms when remembering this process. Using the FOIL method on (-1 + 2i)(-2 + 3i):

• First: Multiply the first terms in each binomial: (-1) * (-2) = 2.
• Outer: Multiply the outer terms in each binomial: (-1) * (3i) = -3i.
• Inner: Multiply the inner terms in each binomial: (2i) * (-2) = -4i.
• Last: Multiply the last terms in each binomial: (2i) * (3i) = 6i^2.

By enumerating each step and summing up the products, one can be assured of accuracy in computation and expansion.

Simplification of Expressions

Simplifying expressions in complex numbers involves combining like terms and replacing powers of i when necessary. After expanding the expression through distributive or FOIL methods, it is essential to reduce it to a simpler form: - Combine like terms in the expression. For imaginary terms such as -3i and -4i, add them to get -7i. - Replace any i^2 terms with -1, owing to the definition of the imaginary unit. Here, 6i^2 is simplified to 6(-1), translating to -6. Finally, add or subtract the real numbers. Updates to the real portion of the expression are straightforward once all imaginary components have been accounted for. Combine the real parts to complete simplification, resulting in a neater form. Ultimately, -4 - 7i is the simplified expression showcasing these simplifications.