Problem 39 - Algebra 2

Question

Simplify. $$ (1-4 i)(2+i) $$

Step-by-Step Solution

Answer

The expression simplifies to \(6 - 7i\).

Step 1: Apply the Distributive Property

To simplify the expression \[(1-4i)(2+i),\]use the distributive property (also called the FOIL method for binomials). Expand the product:\[(1)(2) + (1)(i) - (4i)(2) - (4i)(i).\]

Step 2: Calculate Each Term

Now calculate each term separately:- For \((1)(2)\), the result is \(2\).- For \((1)(i)\), the result is \(i\).- For \((4i)(2)\), the result is \(-8i\).- For \((-4i)(i)\), since \(i^2 = -1\), the result is \(4\).

Step 3: Combine Like Terms

Now, combine all the terms:\[2 + i - 8i + 4.\]Add the real parts and the imaginary parts separately:- Combine the real parts: \(2 + 4 = 6\).- Combine the imaginary parts: \(i - 8i = -7i\).Thus, the expression simplifies to:\[6 - 7i.\]

Key Concepts

Distributive Property

The distributive property is a fundamental mathematical principle that you often use when simplifying algebraic expressions. It allows you to multiply a single term by each term in a polynomial or binomial. This principle is particularly helpful when dealing with expressions that contain variables or complex numbers.

In our example, \[(1 - 4i)(2 + i),\]the distributive property helps us break down the multiplication into more manageable parts. Essentially, it tells us to "distribute" each term from the first binomial to every term in the second binomial. This distributive action can be summarized in the steps:

• Multiply the first term of the first polynomial by each term of the other polynomial.
• Repeat this process with each subsequent term of the first polynomial.

Thus, using the distributive property is like unpacking each term and multiplying them individually, ensuring no part of the expression is left out.

FOIL Method

The FOIL method is a specific application of the distributive property tailored for multiplying two binomials—expressions with two terms each. FOIL stands for First, Outer, Inner, Last, which are the steps to follow.

Here's how it works using the expression \[(1 - 4i)(2 + i):\]

• First: Multiply the first terms in each binomial, \(1 imes 2 = 2.\)
• Outer: Multiply the outer terms,\(1 imes i = i.\)
• Inner: Multiply the inner terms,\(-4i imes 2 = -8i.\)
• Last: Multiply the last terms,\(-4i imes i.\)

This last multiplication involves a special rule for complex numbers, as it includes the imaginary unit. The steps of FOIL allow you to systematically expand the product of the binomials, ensuring all pairs are covered without missing any term. This method simplifies organization in multiplication, especially useful with complex numbers where tracking real and imaginary terms separately is crucial.

Imaginary Unit

The imaginary unit, denoted as \(i,\) is fundamental in the world of complex numbers and mathematics in general. By definition, \(i\) is the square root of \(-1,\) so \(i^2 = -1.\)

This seemingly simple idea allows mathematicians and engineers to solve equations that have no real solutions, greatly expanding the scope of mathematical problems we can tackle. In calculations, recognizing when \(i^2\) occurs helps in simplifying expressions further. For instance, in the multiplication of the expression \((-4i imes i),\) knowing that \(i^2 = -1\) turns this into:\(-4i^2\) which simplifies to \(4\) because \(-4 imes -1 = 4.\)

Hence, when working with the imaginary unit, remember that it often transforms terms significantly, switching between imaginary and real, as it brings negative signs into play. Understanding this property is key to mastering problems involving complex numbers.