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Chapter 1: Problem 56

Find each product. Write the answer in standard form. $$(2+i)^{2}$$

Short Answer

Expert verified
The product is \(3 + 4i\).

Step by step solution

01

Write the expression to be expanded

The given expression is \( (2 + i)^2 \). This can be expanded using the formula for the square of a binomial: \( (a + b)^2 = a^2 + 2ab + b^2 \).
02

Identify the terms

In this expression: \( a = 2 \) and \( b = i \).
03

Expand the expression

Using the formula, \[ (2 + i)^2 = (2)^2 + 2(2)(i) + (i)^2 \].
04

Calculate each term

Calculate each term in the expanded expression: \[ (2)^2 = 4 \] \[ 2(2)(i) = 4i \] \[ (i)^2 = i^2 = -1 \].
05

Sum the terms

Combine all the calculated terms: \[ 4 + 4i - 1 \].
06

Simplify

Combine like terms: \[ 4 - 1 + 4i = 3 + 4i \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers extend the idea of the one-dimensional number line to the complex plane. Here, numbers have two parts: a real part and an imaginary part. For example, in the number \(2 + i\), 2 is the real part, and \(i\) is the imaginary part. The imaginary unit \(i\) is defined by the property \(i^2 = -1\).

  • **Real Part**: The real part of a complex number is a real number. In \(2 + i\), the real part is 2.
  • **Imaginary Part**: This is the value multiplied by \(i\). In \(2 + i\), the imaginary part is 1.
  • **Representation**: A complex number is generally written as \(a + bi\), where \(a\) and \(b\) are real numbers.
Complex numbers make it easier to solve equations that have no real solutions. For instance, the equation \(x^2 + 1 = 0\) has no real solutions because no real number squared gives -1, but it has two complex solutions: \(i\) and \(-i\).
Binomial Expansion
Binomial expansion involves expanding expressions that are raised to a power, represented as \((a + b)^n\). In our exercise, we expand \((2 + i)^2\).
  • **Square of a Binomial**: When expanding \((a + b)^2\), we can use the formula \(a^2 + 2ab + b^2\).
  • **Identify Variables**: For \((2 + i)^2\), let \(a = 2\) and \(b = i\).
  • **Expand**: Plugging these values into the formula gives us \((2)^2 + 2(2)(i) + (i)^2\).
  • **Evaluate Each Term**: Calculate each part separately: \((2)^2 = 4\), \(2(2)(i) = 4i\), and \((i)^2 = -1\).
This stepwise process makes it easy to handle more complex expressions involving binomials.
Standard Form
Writing complex numbers in standard form involves combining the real and imaginary parts. The standard form for complex numbers is \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

  • **Combine Like Terms**: Start with the separately calculated real and imaginary parts. For example, from our previous calculations: \(4 + 4i - 1\).
  • **Simplify**: Combine the real numbers: \(4 - 1 + 4i = 3 + 4i\).
This gives us the standard form. It's useful for easily identifying the real and imaginary parts of the complex number. Standard form is also crucial in performing operations like addition, subtraction, multiplication, and division of complex numbers.

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Most popular questions from this chapter

The manager of a cherry orchard wants to schedule the annual harvest. If the cherries are picked now, the average yield per tree will be \(100 \mathrm{lb},\) and the cherries can be sold for 40 cents per pound. Past experience shows that the yield per tree will increase about 5 lb per week, while the price will decrease about 2 cents per pound per week. How many weeks should the manager wait to get an average revenue of \(\$ 38.40\) per tree?

Simplify each power of i. $$i^{-14}$$

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{2-i}{2+i}$$

Solve each rational inequality. Write each solution set in interval notation. $\frac{1}{x+2} \geq 3$$

Complex numbers are used to describe current I, voltage \(E,\) and impedance \(Z\) (the opposition to current). These three quantities are related by the equation \(E=I Z, \quad\) which is known as Ohm's Law. Thus, if any two of these quantities are known, the third can be found. In each exercise, solve the equation \(E=I Z\) for the remaining value. $$E=57+67 i, Z=9+5 i$$
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