Mathematics

2169 Followers - 222 Articles - 416 Questions and Answers

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 \pi.

Let's begin by simplifying and factoring expressions:


Take a look at: 12x ^ 7+3x ^ 6-10x ^ 5-4x ^ 4+2x ^ 3+5x ^ 2+x-17. 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: 7xw ^ {15}+9xw ^ {15}-2xw ^ {15}=14xw ^ {15}. As you can see, that ugly xw ^ {15} 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.
Photo 18404

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:  4xy+2xw=2x(2y+w). Yeah, since the 2x 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. x ^ 2+x-6 is a good example. Note that: x ^ 2+x-6=x ^ 2+3x-2x-6=x(x+3)-2(x+3)=(x-2)(x+3). 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.  (a+b) ^ 2=a ^ 2+2ab+b ^ 2

2.  (a-b) ^ 2=a ^ 2-2ab+b ^ 2

3.  (a+b)(a-b)=a ^ 2- b ^ 2

These should help get you through the day.

And now, let's move on to quadratic equations

Quadratic equations always look like this: AX ^ 2+BX+C=0, for any numbers A, B and C. Generally though, you can simplify any expression with an x ^ 2 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 AX ^ 2+BX+C=0 into a nice (ax-p)(bx-q)=0 situation. Once there, it's clear that either ax-p=0 or bx-q=0. 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: x=\frac{- B+/- \sqrt{B ^ 2-4AC}}{2A}. 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, \frac{x ^ 6 * x ^ 8}{x ^ {12}}. Well, actually, this expression is equivalent to x ^ 2, 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. (x ^ a) * (x ^ b)=x ^ {a+b} bq. b. \frac{x ^ a}{x ^ b}=x ^ {a-b} bq. c. (x ^ 0)= 1 bq. d. (x ^ a) ^ b=x ^ {a * b} bq. e.  x ^ {\frac{1}{a}}=\sqrt[a]{x}


Now you can quickly see how 4 ^ {\frac{3}{2}} * 2 ^ {-2}= (\sqrt{4}) ^ 3 * 2 ^ {-2}= 2 ^ 3 * 2 ^ {-2}= 2 ^ {3-2}= 2 .

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!

Click Here for Algebra-part-i

Click Here for Algebra-part-ii

Click Here for Algebra-part-iv

Click Here for Algebra-part-v

20 Comments
    jasperb
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    Brenden JasperWed, 11 May 2011 17:46:50 -0000

    thanks for clearing that up i was confused with that symbol

    Post Comments

    Reply to This
    gmat5878
    Vote
    Current Rating
    2
    Rate Up
    Rate Down
    gmat5878Mon, 04 Apr 2011 08:24:28 -0000

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

    Post Comments

    Reply to This
    Shamit123
    Vote
    Current Rating
    0
    Rate Up
    Rate Down
    Shamit123Sun, 01 Mar 2009 16:58:56 -0000

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

    Post Comments

    oLahav
    Rating
    0
    Rate Up
    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.

    Reply to This
    akshay kalambur
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    akshay kalamburSat, 03 Jan 2009 13:26:23 -0000

    real good and this one is very good

    Post Comments

    Reply to This
    rohitv211
    Vote
    Current Rating
    0
    Rate Up
    Rate Down
    rohit varmaTue, 25 Nov 2008 20:42:42 -0000

    nice lesson…..easy to understand. : )

    Post Comments

    Reply to This
    oLahav
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    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.

    Post Comments

    Reply to This
    monali tanna
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    monali tannaMon, 03 Nov 2008 17:12:13 -0000

    it was nice attempt but needs 2 hav more precise explanation

    Post Comments

    Reply to This
    ridhim
    Vote
    Current Rating
    0
    Rate Up
    Rate Down
    Ridhim wadhwaMon, 03 Nov 2008 13:34:52 -0000

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

    Post Comments

    Reply to This
    paarikh
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    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.

    Post Comments

    Hyndavi Bhanu
    Rating
    1
    Rate Up
    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.

    Reply to This
    danish
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    danishTue, 02 Sep 2008 11:12:37 -0000

    send me info

    Post Comments

    Reply to This
    danish
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    danishTue, 02 Sep 2008 11:12:25 -0000

    Post Comments

    Reply to This
    danish
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    danishTue, 02 Sep 2008 11:12:23 -0000

    Post Comments

    Reply to This
    danish
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    danishTue, 02 Sep 2008 11:12:08 -0000

    Post Comments

    Reply to This
    danish
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    danishTue, 02 Sep 2008 11:11:23 -0000

    Nice attempt

    Post Comments

    Reply to This
    mnjkr123
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    mnjkr123Tue, 02 Sep 2008 08:57:36 -0000

    Please contact mnjkr123@hotmail.com for SAT, GMAT lessons

    Post Comments

    Reply to This
    thinkingpad
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    ThinkingpadSun, 31 Aug 2008 07:39:41 -0000

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

    Post Comments

    Reply to This
    magdaaltman
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    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!

    Post Comments

    Reply to This
    babceo
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    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.

    Post Comments

    oLahav
    Rating
    1
    Rate Up
    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.

    Reply to This
    boopathi
    Vote
    Current Rating
    0
    Rate Up
    Rate Down
    boopathiSat, 26 Jul 2008 12:11:43 -0000

    Post Comments

    Reply to This
    Pallabi
    Vote
    Current Rating
    1
    Rate Up
    Rate Down
    Pallabi HuiWed, 11 Jun 2008 09:50:26 -0000

    Very interesting lesson….but I have not got my concepts cleared about the sign ^ in one of the exams

    Post Comments

    oLahav
    Rating
    0
    Rate Up
    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.

    Reply to This

Your Comment

Avatar
Vote
Current Rating
0
Rate Up
Rate Down
Have an account? Log In

Textile is Enabled. View Reference.
Lead_arrow_side

About the Author

oLahav
Name:
About: I don't know how to describe myself... besides, I'm way too biased in this particular topic. What's the point?

Last Updated At Dec 13, 2012
5632 Views

20 Comments factoring quadratic equations exponent laws algebra

Similar Articles

More Lessons