Multiplying Conjugates

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(C)Copyright 2016, C. Burke.

It's sort of the Reverse Lazarus Effect.

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What's a Conjugate?

In Algebra, what is a conjugate? First, it's a noun, not a verb, and it's pronounced something like CON-juh-git, depending on your regional accent, but NOT as con-jyoo-GATE, like a big Language Arts scandal blasted across front pages of the tabloids.

The conjugate of a binomial, an algebraic expression with two terms, is a second binomial with the same terms but the sign between them has changed from plus to minus or minus to plus.

For example, 3x - 7 and 3x + 7 are conjugates.

What makes them interesting? One property of conjugates is to make things GO AWAY, and if there's one thing that Algebra students like is when things go away. And since I refuse to leave, this is the next best thing.

Add, Subtract, Multiply

If you add two conjugates, you double the first term and eliminate the second: (3x - 7) + (3x + 7) = 6x
If you subtract two conjugates, you elimated the first term and double the second: (3x - 7) - (3x + 7) = -14
-- keeping the sign of the term in the first binomial.

If you multiply them, something interesting happens:

You get a Difference of Squares. That is, the square of the first term minus the square of the second term. When you do the Distributive Property, you should get two more terms -- and don't you forget that! -- but in this case, those terms will cancel out! (-xy) + (xy) = 0.
(3x - 7)(3x + 7) = 9x² + 21x - 21x - 49 = 9x² - 49.

This can be useful not just for multiplying binomials, but for multiplying actual, honest-to-goodness, Real numbers, too!
Take, for example, (16) X (24). Not really easy to do in your head, but if you split the difference, you can see that it is the same as (20 - 4)(20 + 4).

Voila! The answer is 384! Wasn't that easy? Isn't this the greatest trick?

Nah, it's not. Just Kidding, really.

Just use a calculator. Seriously. No one really wants to square "bad" numbers in their head and then subtract them! But sometimes, it's kinda cool and you can impress your friends if you carefully pick your numbers!

Radicals!

BUT WAIT, THERE'S MORE!

Suppose you have a binomial where one of the terms is a radical number. Wouldn't you like that to go away, too? Well you can! Just multiply it by the conjugate.

I know, I know what you're thinking. You're thinking, "Yeah, that's okay, Mr. Burke. I'm cool with the radical. I'll just leave it alone!"

That's nice that you're cool with it, but you can't leave it alone. Suppose you have two divided by (6 plus radical 7). If there's a radical in the denominator of a fraction, it has to go away. That's just the rule. We're going to "simplify" it by multiplying both the numerator and the denominator of the fraction the conjugate, like this:

Isn't that so much better? It is, isn't it? Worth it, right? Right?

Imaginary Numbers

The same way that conjugates work for radical numbers, they can work with imaginary numbers.

If you have 3 + 4i, for example, in the bottom of a fraction again, you can make it real by multiplying by the conjugate, 3 - 4i.

Using our rule from about (3 - 4i)(3 + 4i) = 9 + 16 = 25, which looks suspiciously like a part of a Pythagorean Theorem problem -- but that's for another night.