Remember that you can distribute multiplication over addition? You can also distribute exponents over multiplication.
Here’s how and why it works:
And here it is written out “long ways” for an algebra problem. Remember, you can use the short version. I do the videos with the long version to help you remember why it works.
Have any questions? Post them in the comments!
We learn to factor polynomials so we can solve for x. Stuff like
3x2 + 2x – 1
can be easily factored into
(3x – 1)(x + 1)
But some things aren’t so easy. And some things are just down right a pain in the bottom. Like this one:
18x2 + 189x + 490
“Just shoot me,” you might be thinking. But cool your jets, cowboy. Here’s the first tool you need to never factor polynomials again:
I know – who wants to memorize that formula? But wait. If you use that one formula that you can memorize, you never have to factor polynomials again. Watch:
So the factoring isn’t bad on that one, right? How about this one:
And how about if you come across one of these. Wouldn’t it be nice to get rid of all guesswork:
This has some disadvantages, of course. You see that there’s a lot more arithmetic. And there’s some things to look out for. But if you hate trial and error (like I do), then you might be willing to take the bad with that good.
Whatcha think?
One of the common errors in algebra is canceling bits in fractions that can’t be canceled. To remember the rule, though, you don’t have to memorize anything. You only have to think of a plain arithmetic problem to compare it.
To help you remember what’s allowed with problem:
think about this simpler problem:
Like this:
And just to keep you straight, here are the requirements (and a couple of examples) for the fancy <cue music> “distributive property.”
What do you think of this explanation? Post your thoughts in the comments.
Kellie writes: “I am told to find the nth number in a pattern. What the heck does nth mean?”
So I’ll first answer the question directly. Then we can talk about what it means to the rest of the problems.
One more piece of the puzzle before we’re on to solve problems. I’ll bet you saw something in there about n-1 or n+1. Here’s a bit about that…
Now let’s get into writing the nth term of a sequence using the one before it. Remember the nth term is the number in the list if you count up to whatever you’ve labeled n.
In the previous video, we would have written it in fancy terms like this:
Once more, just to get the hang of it:
Again, this one would have been written:
Aha! But what if you don’t know the number right before the one you want? Watch this:
Still have questions? Post them in the comments.
Thanks Kellie, for your question!
Complex numbers are made from one part real number and one part imaginary number. That’s one heck of a mixture.
Anytime you mix things, you take the complicated to, well, the complex. (Goodness I’m punny today!)
We use conjugates in the manipulation of these things. First it’s important to understand what a conjugate is…
It’s just a pixie-dust quick-switch sleight of hand thing, right? Pretty much. Can you legally do that? Well, no. Clearly 4+8 is not the same as 4-8. Anymore than 12 is the same as -4. Duh.
But you’ll see in this next video that you’re not really jumping up and trying to change the value of things…
What’s really happening is that you’re inventing something to use as a “clever form of one” – the big fat one.
And here’s one more example. Don’t forget to comment if you’re still confused.
Completing the square isn’t as hard as everyone thinks. And there’s no reason to memorize a bunch of strange pixie dust magic math rules. Here’s the way to see it for what it is.
Here’s another example:
This last one involves fractions:
Have questions? Post them in the comments.
Do you ever wonder why teachers start a problem the way they do?
There are four reasonable ways to start solving an equation like:
3x + 2 = 4x – 1
On this site I will always manipulate and move things around so that the equation we’re solving looks like:
(the three colors represent three different numbers)
I have done it here:
Have questions? Post them in the comments.