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Chapter 3: Problem 69

Multiply as indicated. Write each product in standard form. $$3 i(2-i)^{2}$$

Short Answer

Expert verified
The product in standard form is \(12 + 9i\).

Step by step solution

01

Expand the Exponent

First, let's expand the square of the binomial \((2-i)^2\). This can be done by using the formula for squaring a binomial, which is \((a-b)^2 = a^2 - 2ab + b^2\).For \(a = 2\) and \(b = i\), we have:\[(2-i)^2 = 2^2 - 2 \cdot 2 \cdot i + i^2\] \[= 4 - 4i + i^2\]Since \(i^2 = -1\), substitute \(i^2\) with \(-1\):\[= 4 - 4i - 1\]Simplify the expression:\[= 3 - 4i\]
02

Multiply by the Scalar

Now, we need to multiply the result from Step 1 by \(3i\), as indicated in the original expression. Thus, we are multiplying:\[3i(3 - 4i)\]Distribute \(3i\) across each term inside the parentheses:\(3i \times 3 = 9i\)\(3i \times (- 4i) = -12i^2\)Since \(i^2 = -1\), this becomes:\(-12i^2 = -12(-1) = 12\)Thus, the product is:\[= 12 + 9i\]
03

Write in Standard Form

The standard form for complex numbers is \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.From Step 2, we have:\[12 + 9i\]This is already in standard form with the real part \(12\) and the imaginary part \(9i\).

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Key Concepts

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

Binomial Expansion
The Binomial Expansion is a powerful mathematical tool, allowing us to expand expressions raised to a power. It is super helpful in solving polynomial equations. In our exercise, we focused on squaring a binomial, which involves using the formula
  • \((a-b)^2 = a^2 - 2ab + b^2\)
For the binomial \((2-i)^2\), we substitute
  • \(a = 2\)
  • \(b = i\)
This results in:
  • \((2-i)^2 = 2^2 - 2 imes 2 imes i + i^2\)
By simplifying, you get:
  • \(= 4 - 4i + i^2\)
Recognizing that \(i^2 = -1\) simplifies it further, thus giving:
  • \(= 4 - 4i - 1 = 3 - 4i\)
This expansion is a vital skill in dealing with complex numbers.
Standard Form
Writing complex numbers in Standard Form, \(a + bi\), is essential for easy readability and comparison.
This form has two parts:
  • \(a\) is the real part.
  • \(bi\) is the imaginary part.
In our solution, after expanding and multiplying, we obtained:
  • \(= 12 + 9i\)
Here:
  • \(12\) is the real part.
  • \(9i\) is the imaginary part.
Arranging in this form ensures that the complex number is clearly structured, aiding in further calculations or graphing in the complex plane.
Imaginary Unit
The Imaginary Unit, represented as \(i\), is a fundamental concept in complex numbers. It is defined with the property:
  • \(i^2 = -1\)
This unique property allows us to work with numbers that extend beyond the real number line.
For example, in our exercise, when expanding and simplifying \((2-i)^2\), the term \(i^2 = -1\) allowed us to simplify the expression.
When multiplying with \(3i\), using \(i^2 = -1\) turned \(-12i^2\) into \(12\).
This transformation is crucial in finding the solution in a form that is easy to understand and work with further.
Understanding \(i\) is key to mastering complex numbers, as it forms the foundation of many advanced topics.

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Most popular questions from this chapter

Use the concepts of this section. Suppose that \(k, a, b,\) and \(c\) are real numbers, \(a \neq 0,\) and a polynomial function \(P(x)\) may be expressed in factored form as \((x-k)\left(a x^{2}+b x+c\right)\). (a) What is the degree of \(P ?\) (b) What are the possible numbers of distinct real zeros of \(P ?\) (c) What are the possible numbers of nonreal complex zeros of \(P ?\) (d) Use the discriminant to explain how to determine the number and type of zeros of \(P\).

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=x^{3}-7 x^{2}+13 x-3 ; 3$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$s=\frac{1}{2} g t^{2} \quad \text { for } t$$

For each polynomial function, do the following in order. (a) Use Descartes' rule of signs to find the possible number of positive and negative real zeros. (b) Use the rational zeros theorem to determine the possible rational zeros of the function. (c) Find the rational zeros, if any. (d) Find all other real zeros, if any. (e) Find any other nonreal complex zeros, if any. (f) Find the \(x\) -intercepts of the graph, if any. (g) Find the \(y\) -intercept of the graph. (h) Use synthetic division to find \(P(4),\) and give the coordinates of the corresponding point on the graph. (i) Determine the end behavior of the graph. (i) Sketch the graph. (You may wish to support your answer with a calculator graph.) $$P(x)=-2 x^{4}-x^{3}+x+2$$

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=x^{5}-3 x^{3}+x+2\); no real zero less than \(-3\)
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