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Chapter 3: Problem 20

Find each product. $$[x-(3-i)][x-(3+i)]$$

Short Answer

Expert verified
The product is \[ x^2 - 6x + 10 \].

Step by step solution

01

- Write the Expression with Conjugates

The given expression is \[ x-(3-i) \] and \[ x-(3+i) \]. Recognize that these two binomials are complex conjugates.
02

- Apply the Difference of Squares Formula

When multiplying conjugates \[ (a+b)(a-b) = a^2 - b^2 \]. Here, represent \[ 3-i \] as \[ a \] and \[ 3+i \] as \[ b \]. Substitute \[ (x-3+i)(x-3-i) \] into the formula to get \[ (x-3)^2 - (i)^2 \].
03

- Simplify the Squares

First, compute the square of \[ x-3 \]: \[ (x-3)^2 = x^2 - 6x + 9 \]. Then compute \[ i^2 = -1 \].
04

- Substitute and Simplify

Replace in the equation: \[ (x-3)^2 - (-1) = x^2 - 6x + 9 + 1 = x^2 - 6x + 10 \].

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Key Concepts

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

complex numbers
Complex numbers are numbers that have both a real part and an imaginary part. The general form of a complex number is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, with \(i\) being the imaginary unit. The imaginary unit \(i\) is defined by the property \(i^2 = -1\).
For example, in the complex number \(3 - i\), 3 is the real part and \(-i\) is the imaginary part.
Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. For instance, \(3 - i\) and \(3 + i\) are complex conjugates. When we multiply complex conjugates, interesting and simplifying patterns emerge.
difference of squares
The difference of squares is a mathematical pattern that appears when we multiply two binomials of the form \(a + b\) and \(a - b\). The product results in the equation \(a^2 - b^2\).
In the context of complex conjugates, this formula becomes very useful. Consider the expression \[ (x - (3 - i))(x - (3 + i)) \]. These binomials are complex conjugates.
When applying the difference of squares formula, we treat \((x - 3)\) as 鈥榓鈥� and \(i\) as 鈥榖鈥�. Therefore:
  • \[(x - 3)^2 - (i)^2 \]
  • Simplify \((x - 3)^2\)
  • Recognize \(i^2 = -1\)
This approach helps to simplify the multiplication of two complex conjugate binomials effectively.
simplifying algebraic expressions
Simplifying algebraic expressions is the process of making an expression more concise and easier to work with. This typically involves combining like terms, factoring, and eliminating unnecessary parts.
For the given problem, we start by simplifying \((x - 3)^2\). This is done by applying the square of a binomial formula:
  • \[(x - 3)^2 = x^2 - 6x + 9 \]
Then, we address the term \(i^2\), recalling that \(i^2 = -1\) and simplifies to add 1:
  • \[(x - 3)^2 - (i)^2 = x^2 - 6x + 9 - (-1) = x^2 - 6x + 9 + 1 = x^2 - 6x + 10 \]
This process transforms a complex expression into a more manageable polynomial form.

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Most popular questions from this chapter

Use Descartes's rule of signs to discuss the possibilities for the roots of the equation \(-x^{4}-6 x^{2}-3 x+9=0\)

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