Question

which pair of complex factors results in a real number product?

(3i)(3i)

(3i)(-3)

(5 + 2i)(5 - 2i)

(4 + 5i)(4 + 5i)

Explanation:

Based on the image provided, the question asks:
"Which pair of complex factors yields a real number product?"

The options are:

1. \((2i)(-2)\)
2. \((3i)(-3)\)
3. \((5 + 2i)(5 - 2i)\)
4. \((4 + 5i)(4 + 5i)\)

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🆕 New Concept Discovered: Complex Conjugates
Multiplying a complex number by its conjugate always results in a real number.

Step 1: Understand the Complex Conjugate Rule

A complex number has the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i = \sqrt{-1}\). Its complex conjugate is \(a - bi\) (the sign of the imaginary part is flipped).

When you multiply a complex number by its conjugate, the imaginary terms cancel out, leaving a purely real number:

$$ (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2i^2 $$

Since \(i^2 = -1\):

$$ a^2 - b^2(-1) = a^2 + b^2 $$

Because \(a\) and \(b\) are real numbers, \(a^2 + b^2\) must be a real number.

Step 2: Evaluate the Options

• Option 1: \((2i)(-2) = -4i\) (This is an imaginary number)
• Option 2: \((3i)(-3) = -9i\) (This is an imaginary number)
• Option 3: \((5 + 2i)(5 - 2i)\)

These two factors are complex conjugates of each other. Let's multiply them:

$$ (5 + 2i)(5 - 2i) = 5^2 - (2i)^2 = 25 - 4i^2 = 25 - 4(-1) = 25 + 4 = 29 $$

Since \(29\) has no imaginary part, this product is a real number.

• Option 4: \((4 + 5i)(4 + 5i) = (4 + 5i)^2 = 16 + 40i + 25i^2 = 16 + 40i - 25 = -9 + 40i\) (This is a complex number with an imaginary part)

Answer:

\((5 + 2i)(5 - 2i)\)