Question

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

\\(15(-15i)\\)
\\(3i(3i)\\)
\\((8 + 2i)(-8 - 2i)\\)
\\((4 - 7i)(4 - 7i)\\)

Explanation:

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

Step 1: Understand the rule for a real-number product

When we multiply two complex numbers, the result is generally another complex number. However, if we multiply a complex number \(a + bi\) by its complex conjugate \(a - bi\), the imaginary parts cancel out:

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

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

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

Because \(a\) and \(b\) are real numbers, \(a^2 + b^2\) is always a real number. Therefore, we need to find the pair of factors that are complex conjugates of each other (same real parts, opposite imaginary parts).

Step 2: Analyze the given options

Let's test the options to see which one represents a pair of complex conjugates:

• Option 1: \(15(-15)\)

These are real numbers, not a pair of complex factors with imaginary parts.

• Option 2: \(3(3i)\)

Multiplying these gives:

$$ 3 \cdot 3i = 9i $$

This is an imaginary number, not a real number.

• Option 3: \((8 + 2i)(-8 - 2i)\)

Let's multiply them:

$$ (8 + 2i)(-8 - 2i) = -64 - 16i - 16i - 4i^2 = -64 - 32i - 4(-1) = -60 - 32i $$

This contains an imaginary term (\(-32i\)), so it is not a real number.

• Option 4: \((4 - 7i)(4 + 7i)\)

These are complex conjugates because they have the same real part (\(4\)) and opposite imaginary parts (\(-7i\) and \(+7i\)). Let's multiply them:

$$ (4 - 7i)(4 + 7i) = 4^2 - (7i)^2 = 16 - 49i^2 = 16 - 49(-1) = 16 + 49 = 65 $$

Since \(65\) is a real number, this pair results in a real-number product.

Answer:

The correct option is the fourth one:
\((4 - 7i)(4 + 7i)\)