Real Number Product: Which Complex Factor Pair Works?
Hey guys! Let's dive into the fascinating world of complex numbers and figure out which pair, when multiplied, gives us a real number. This might sound a bit tricky, but trust me, it's super cool once you get the hang of it. We're going to break down each option, so you'll not only get the answer but also understand why it's the answer. Get ready to explore the magic of complex conjugates and how they play a key role in this mathematical puzzle!
Understanding Complex Numbers and Real Number Products
Before we jump into the options, let's quickly recap what complex numbers are. A complex number is basically a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i² = -1). The a part is the real part, and the bi part is the imaginary part. When we talk about a real number product, we mean that when we multiply two complex numbers, the imaginary part cancels out, leaving us with a regular, real number.
The key to achieving a real number product often lies in something called complex conjugates. The conjugate of a complex number a + bi is a - bi. When you multiply a complex number by its conjugate, the imaginary terms eliminate each other, resulting in a real number. This happens because of the difference of squares pattern: (a + bi)(a - bi) = a² - (bi)² = a² + b². Notice how the i² becomes -1, which cancels out the negative sign, leaving us with a sum of squares – a real number!
So, with this knowledge in our arsenal, we can approach the given options and see which pair of complex factors fits the bill. We're essentially looking for a pair that are either complex conjugates of each other or will simplify in a way that the imaginary part disappears upon multiplication. Keep this in mind as we go through each option. Remember, math isn't about memorizing formulas, it's about understanding the underlying concepts. By grasping the concept of complex conjugates and how they lead to real number products, you'll be able to tackle similar problems with confidence!
Analyzing the Options
Let's break down each option step by step to see which pair of complex factors yields a real number product.
A. 15(-15i)
In this option, we have 15 multiplied by -15i. This is a straightforward multiplication. Let's perform the calculation:
15 * (-15i) = -225i
The result is -225i, which is purely imaginary. There's no real part here, so this pair does not result in a real number product. The imaginary unit i is still present, indicating that the result is on the imaginary axis of the complex plane. Therefore, option A is not the correct answer.
B. 3i(1-3i)
Here, we have 3i multiplied by the complex number (1 - 3i). We need to distribute the 3i across the terms inside the parentheses:
3i * (1 - 3i) = 3i - 9i²
Remember that i² = -1, so we can substitute that in:
3i - 9*(-1) = 3i + 9
This simplifies to 9 + 3i, which is a complex number with both a real part (9) and an imaginary part (3i). Since the imaginary part doesn't cancel out, this pair also does not result in a real number product. So, option B is not our answer.
C. (8+20i)(-8-20i)
This option presents us with the product of two complex numbers: (8 + 20i) and (-8 - 20i). Notice something interesting here? The second complex number is the negative of the first. Let's multiply them using the distributive property (also known as FOIL):
(8 + 20i) * (-8 - 20i) = 8*(-8) + 8*(-20i) + 20i(-8) + 20i(-20i)
= -64 - 160i - 160i - 400i²
Now, let's simplify, remembering that i² = -1:
= -64 - 320i + 400
= 336 - 320i
We end up with 336 - 320i, which is still a complex number with a non-zero imaginary part (-320i). Thus, this pair does not result in a real number product either. Option C is not the correct choice.
D. (4+7i)(4-7i)
Finally, let's examine option D: (4 + 7i) multiplied by (4 - 7i). Take a close look – do you notice anything special about these two complex numbers? They are complex conjugates of each other! This is a major clue. When we multiply complex conjugates, we expect the imaginary terms to cancel out. Let's perform the multiplication:
(4 + 7i) * (4 - 7i) = 44 + 4(-7i) + 7i4 + 7i(-7i)
= 16 - 28i + 28i - 49i²
Notice how the -28i and +28i terms cancel each other out. Now, let's substitute i² = -1:
= 16 - 49*(-1)
= 16 + 49
= 65
The result is 65, which is a real number! The imaginary part has completely vanished, leaving us with a real number product. Therefore, option D is the correct answer.
The Verdict: Option D is the Winner!
After carefully analyzing each option, we've discovered that option D, (4 + 7i)(4 - 7i), is the only pair of complex factors that results in a real number product. This is because the two factors are complex conjugates of each other. When complex conjugates are multiplied, the imaginary terms cancel out, leaving us with a real number. Remember this key concept, guys! It's super helpful when dealing with complex numbers.
So, there you have it! We've not only found the answer but also understood the underlying principle behind it. Keep practicing, and you'll become a pro at complex number problems in no time!