Problem 63 - Algebra 2

Question

Simplify. $$ (2+i)(1+2 i)(3-4 i) $$

Step-by-Step Solution

Answer

The expression simplifies to \(12 + 9i\).

Step 1: Distribute the First Two Complex Numbers

First, we'll multiply the first two complex numbers \((2+i)(1+2i)\). Apply the distributive property: \[(2+i)(1+2i) = 2(1) + 2(2i) + i(1) + i(2i)\]Simplifying, we get:\[= 2 + 4i + i + 2i^2\]Since \(i^2 = -1\), replace \(2i^2\) with \(-2\):\[= 2 + 4i + i - 2\]Combine like terms:\[= 3i\]

Step 2: Multiply the Result by the Third Complex Number

Now, take the result from step 1, which is \(3i\), and multiply it by the third complex number \(3 - 4i\):\[(3i)(3 - 4i) = 3i(3) + 3i(-4i)\]Simplify each term individually:\[= 9i - 12i^2\]Since \(i^2 = -1\), replace \(-12i^2\) with \(12\):\[= 9i + 12\]

Step 3: Write the Final Result

The expression \((2+i)(1+2i)(3-4i)\) simplifies to the complex number \(12 + 9i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept which helps us in multiplying two binomials, such as two complex numbers. Essentially, we distribute each term in the first binomial across each term in the second binomial.
For instance, when we multiply \( (2+i)(1+2i) \), we apply the distributive property by multiplying each term in the first binomial by each term in the second binomial: \[ (2+i)(1+2i) = 2(1) + 2(2i) + i(1) + i(2i) \]
This is akin to expanding the two numbers into separate parts, to ensure that no part is left unmultiplied.

• Multiply 2 and 1, yielding 2.
• Next, multiply 2 by \(2i\), resulting in \(4i\).
• Continue with \(i\) and 1, giving another \(i\).
• Finally, multiply \(i\) by \(2i\), which results in \(2i^2\).

Upon simplifying, we combine all these products for our equation: \(2 + 4i + i + 2i^2\).

i squared equals -1

One of the most intriguing aspects of complex numbers is the imaginary unit \(i\), which is defined such that \(i^2 = -1\). This property is crucial when simplifying expressions that include imaginary numbers.
In the context of our problem, we faced terms like \(2i^2\) and \(-12i^2\). Understanding the property of \(i\), we know that:

• \(2i^2\) becomes \(-2\) because \(i^2 = -1\).
• Likewise, \(-12i^2\) simplifies to \(12\).

This unique property turns an otherwise complex issue into a simple arithmetic operation by converting any \(i^2\) term into a negative real number.

Simplifying Expressions

Simplifying expressions involving complex numbers often requires combining like terms and converting imaginary units when possible. We utilize the rules of arithmetic and the special properties of \(i\) for this task.
As seen in our example, after initial distribution and substitution of \(i^2 = -1\), combining like terms becomes straightforward.

• From Step 1, we derived the expression \((2+i)(1+2i) = 2 + 4i + i - 2\).
• Here, combining like terms efficiently gave us \(3i\) by canceling out the real numbers (2 and -2) and adding the imaginary ones (4i and i).

Finally, proceeding to Step 2, our simplified result \(3i\) was multiplied with \(3-4i\) which resulted in combined new terms: \(9i + 12\).
This whole process underscores how recognizing and applying basic mathematical rules alongside specific properties of complex numbers can make simplifying such expressions manageable.