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# Basics of Algebra: Part III

by Oren Lahav

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.

 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: . As you can see, that ugly thing is common to all 3 parts of the expression, so we can make the whole thing become one by adding or subtracting the proper coefficients.

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.

 And now, let's move on to quadratic equations Quadratic equations always look like this: , for any numbers A, B and C. Generally though, you can simplify any expression with an to this form. Once in this form, there are 2 ways to solve this type of equation:

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.

%{font-family:verdana;color:BLUE; font-size:16px}*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:

bq. a. bq. b. bq. c. bq. d. bq. e.

Now you can quickly see how .

 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.

Thanks for reading this Welcome to Algebra Lesson!

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Brenden JasperWed, 11 May 2011 17:46:50 -0000

thanks for clearing that up i was confused with that symbol

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gmat5878Mon, 04 Apr 2011 08:24:28 -0000

thanks Oren Lahav for your very nice explanation. Wishing you all the best.

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Shamit123Sun, 01 Mar 2009 16:58:56 -0000

Hi Pallabi,
This is Shamit - Can you please explain the meaning of ^

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Oren LahavMon, 02 Mar 2009 14:49:41 -0000

The sign ^ is usually used to represent a power on the computer. For example, 3 ^ 2 = 9 and 5 ^ 3 = 125.

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akshay kalamburSat, 03 Jan 2009 13:26:23 -0000

real good and this one is very good

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rohit varmaTue, 25 Nov 2008 20:42:42 -0000

nice lessonâ€¦..easy to understand. : )

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Oren LahavTue, 04 Nov 2008 14:52:27 -0000

You can find lots of practice-specific lessons here, like the algebra problems go-through lesson.

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monali tannaMon, 03 Nov 2008 17:12:13 -0000

it was nice attempt but needs 2 hav more precise explanation

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Ridhim wadhwaMon, 03 Nov 2008 13:34:52 -0000

hey it me thnk tht math is not as tough as i use 2 thnk

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paarikhWed, 22 Oct 2008 11:05:54 -0000

Very Helpful, i was finding it difficult get to terms with mathematics. Since i got into business in 2000, it was like starting afresh with Gmat preparations, all lessons are very helpful.

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Hyndavi BhanuThu, 02 Feb 2012 13:33:17 -0000

but how did you did all that?
i cant just understand anything how to proceed in this website.

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danishTue, 02 Sep 2008 11:12:37 -0000

send me info

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danishTue, 02 Sep 2008 11:12:25 -0000

send me

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danishTue, 02 Sep 2008 11:12:23 -0000

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danishTue, 02 Sep 2008 11:12:08 -0000

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danishTue, 02 Sep 2008 11:11:23 -0000

Nice attempt

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mnjkr123Tue, 02 Sep 2008 08:57:36 -0000

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ThinkingpadSun, 31 Aug 2008 07:39:41 -0000

thank u very much dear oLahav for sach a great effort. thanks again.

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Fri, 22 Aug 2008 22:06:07 -0000

I just want to send oLahav my sincere thanks for putting up these clear and helpful lessons. I haven't done any math since 9th grade (some few eons ago) and this is making the reentry enjoyable!

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Chris BabowalFri, 08 Aug 2008 16:12:37 -0000

This is a nice lesson. It may not be as basic as some beginning students may need. To reach more student you might want to consider using easier English for students who are English language learners. Overall this is a major thumbs up lesson.

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Oren LahavFri, 08 Aug 2008 16:14:46 -0000

Thanks for the comment! It's true that I haven't considered the English level of readers, I'll try to take it into account in future lessons.

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boopathiSat, 26 Jul 2008 12:11:43 -0000

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Pallabi HuiWed, 11 Jun 2008 09:50:26 -0000

Very interesting lessonâ€¦.but I have not got my concepts cleared about the sign ^ in one of the exams

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Oren LahavWed, 11 Jun 2008 13:35:23 -0000

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.

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