Algebra, part II- factoring, quadratic equations, exponent laws
This series of lessons is
designed to help you learn, or review, the fundamentals of algebra.
In this lesson we move on to factoring and simplifying expressions,
solving quads and dealing with exponents.
Algebra isn't as
scary as some people tend to think. Up to know we've dealt with
super-basic algebra. What's coming up next is a bit more
challenging, but like all math, practice will make this as easy as
.
Let's
begin by simplifying and factoring
expressions:
Take a look at:
. Are you
freaking out yet? Ok, we won't deal with that one, but in short,
this sort of thing isn't such a big monster. There are ways, nice
and easy ways, of making this sort of beast become a cute little
poodle. Metaphorically speaking.
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Simplification
means just
that- simplifying huge expressions into nicer ones. Note that this
isn't solving equations (clearly, since there's no = sign, not even
a < or > sign), so you can't just divide everything by
something or subtract something else to make the thing you're
looking at look nicer. What can we do? One of the basic things we can do is to collect like terms. Illustrating this using a simple example: |
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The more important thing
though is factoring.
Like my math
teacher used to say, if you're stuck on an ugly problem and feel
like saying the F word, add a "tor" to the end of it and you get
"factor". (If you don't get this joke, ask me later). Another
simple illustration with an example: . Yeah, since the
is common, we pulled it out,
and got a much nicer thing in exchange. That's the basis of
factoring- find common elements and pull them out.
An immediate and important
thing to do is learn how to factor quadratics. Quadratic
expressions are expressions with 1 variable, where the variable is
raised to the power of 2. is a good
example. Note that:
. Not so
scary now, is it? Practicing will make you expert at factoring this
and other expressions.
Before we go on, a few nice tricks:
Some expressions require you to expand, the opposite of factor. There are a few simple expansion tricks worth remembering:
1.
2.
3.
These should help get you through the day.
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And
now, let's move on to quadratic equations Quadratic equations always look like this: |
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1.
Factoring: Remember this? We can
sometimes turn a nasty
into a nice
situation. Once there, it's clear that either
or
. These
you can solve easily. Note that you'll get 2 possible solutions-
that's ok, that's what should happen most often.
2.
The quadratic formula is the second way of
solving quads. It always works, no matter what, but it can give you
nasty results and it's not as fast. The formula goes like this:
. Note that the +/-
thing means you have to do it twice, once with a + and once with a
- . This will again give you 2 answers. We'll come back to this
formula later on in life to understand complex numbers.
This could be worse,
right? Say, . Well, actually, this expression
is equivalent to
, but to get there you
need to look at exponents and their laws.
Exponent laws- even exponents can be made simple
Yes, believe it or not,
it's true. Here is a short, exhaustive list of the rules you can
use to simplify exponents:
a.
b.
c.
d.
e.
Now you can
quickly see how .
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So, with
everything we learned in this lesson, we can clean up some large,
messy expressions into nice, simple ones, and then solve them if
they're in equations. See, math isn't such an awful thing after
all. Next time, we'll get into algebra that has to do with number theory, with some stuff about primes, complex numbers and more. |
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Thanks for reading this Welcome to Algebra Lesson!
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Post Comments
oLahav said – Wed, 11 Jun 2008 13:35:23 -0000 ( Flag Edit Link )
The sign ^ can be a confusing one if you’ve never seen it before. It represents exponents, so for example 2 ^ 2 is 2 to the power of 2, which is 4.
You should watch out for the brackets here: 2 ^ 2 + 1 is 4 + 1=5, but 2 ^ (2+1) is 2 ^ 3=8.
I hope this clears things up, if not start a discussion about mathematical signs in computer language, I’m sure it’ll help lots of people.