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Chapter 3: Problem 67

Multiply as indicated. Write each product in standard form. $$i(3-4 i)(3+4 i)$$

Short Answer

Expert verified
The product is \(25i\).

Step by step solution

01

Identify the Expression

We want to multiply the complex numbers \( (3 - 4i) \) and \( (3 + 4i) \) as indicated and then multiply the result by \( i \). This requires applying the distributive property and rearranging the result in standard form \( a + bi \).
02

Apply the Difference of Squares

Recognize that \((3 - 4i)(3 + 4i)\) is a difference of squares, given by \((a - b)(a + b) = a^2 - b^2\). Here, \(a = 3\) and \(b = 4i\).
03

Calculate the Conjugate Product

Apply the identity \((3 - 4i)(3 + 4i) = 3^2 - (4i)^2\). This results in \(9 - (16i^2)\). Since \(i^2 = -1\), \((16i^2) = 16(-1) = -16\). Substitute to get \(9 + 16 = 25\).
04

Multiply by i

Now, multiply this real number result, 25, by \(i\): \[i imes 25 = 25i\].
05

Write in Standard Form

The expression is now \(0 + 25i\), which is already in standard form \(a + bi\) with \(a = 0\) and \(b = 25\).

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Key Concepts

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

Multiplication of Complex Numbers
Multiplying complex numbers can be a bit tricky, but it's all about using the distributive property effectively and understanding the nature of imaginary numbers. When you multiply two complex numbers, such as \((a + bi)(c + di)\), you'll follow these steps:
  • Use the distributive property to expand the product: \((a + bi)(c + di) = ac + adi + bci + bdi^2\).
  • Remember that \(i^2 = -1\). This is crucial because it allows you to simplify terms involving \(i^2\) to real numbers.
  • Combine like terms, specifically separating real numbers from imaginary numbers.
The given exercise involved multiplying two conjugates, \((3-4i)\) and \((3+4i)\), and then multiplying the result by \(i\). This is a great opportunity to practice these core multiplication principles.
Standard Form of Complex Numbers
Complex numbers are typically written in what's known as "standard form," which is \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This format makes it easy to identify and work with the real and imaginary parts of a complex number.

After performing operations on complex numbers, such as addition, subtraction, multiplication, or division, it is a standard practice to express the result in this form. In the exercise, the final product after multiplying was given as \(0 + 25i\). This expression is already in standard form:
  • The real part, \(a\), is 0.
  • The imaginary part, \(b\), is 25.
Writing complex numbers in standard form will help you better analyze and apply them in various mathematical contexts.
Difference of Squares
The difference of squares is a powerful algebraic identity that simplifies expressions and calculations in mathematics. This identity is expressed as \((a - b)(a + b) = a^2 - b^2\). It is particularly useful when working with conjugate pairs of complex numbers.

In the provided exercise, this concept was applied to multiply \((3 - 4i)\) and \((3 + 4i)\). Recognizing them as a difference of squares, the calculation was simplified to \(3^2 - (4i)^2\):
  • Calculate \(3^2 = 9\).
  • Calculate \((4i)^2 = 16i^2\), which simplifies to \(-16\) because \(i^2 = -1\).
  • Simplification leads to \(9 - (-16) = 9 + 16 = 25\).
This clever use of the difference of squares can simplify complex number calculations considerably, highlighting the elegance of algebraic identities.

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Most popular questions from this chapter

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then, use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=5 x^{4}+3 x^{2}+2 x-9$$

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=3 x^{4}+2 x^{3}-4 x^{2}+x-1\); no real zero greater than 1

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=x^{4}+3 x^{3}-3 x^{2}-11 x-6 \\\&=(x+3)(x+1)^{2}(x-2)\end{aligned}$$

Use the rational zeros theorem to factor \(P(x)\). $$P(x)=12 x^{3}+20 x^{2}-x-6$$

Use the concepts of this section. Determine whether the description of the polynomial function \(P(x)\) with real coefficients is possible or not possible. (a) \(P(x)\) is of degree 3 and has zeros of \(1,2,\) and \(1+i\). (b) \(P(x)\) is of degree 4 and has four nonreal complex zeros. (c) \(P(x)\) is of degree 5 and \(-6\) is a zero of multiplicity 6. (d) \(P(x)\) has \(1+2 i\) as a zero of multiplicity 2.
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