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Basics of Algebra: Part III

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.

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. (x ^ a) * (x ^ b)=x ^ {a+b}

b. \frac{x ^ a}{x ^ b}=x ^ {a-b}

c. (x ^ 0)= 1

d. (x ^ a) ^ b=x ^ {a * b}

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!
Pallabi
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Pallabi said:

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

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  • Posted 6 months ago.
oLahav
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oLahav said in response to:
Pallabi
Pallabi’s post:
Citation Body

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

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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  • Posted 6 months ago.
boopathi
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boopathi said:

good

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  • Posted 5 months ago.
babceo
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babceo said:

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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  • Posted 5 months ago.
oLahav
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oLahav said in response to:
babceo
babceo’s post:
Citation Body

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.

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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  • Posted 5 months ago.
magdaaltman
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magdaaltman said:

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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  • Posted 4 months ago.
thinkingpad
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thinkingpad said:

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

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  • Posted 4 months ago.
mnjkr123
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mnjkr123 said:

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

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  • Posted 4 months ago.
danish
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danish said:

Nice attempt

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  • Posted 4 months ago.
danish
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danish said:

Please

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  • Posted 4 months ago.
danish
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danish said:

send

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danish
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danish said:

send me

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danish
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danish said:

send me info

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  • Posted 4 months ago.
paarikh
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paarikh said:

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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  • Posted 2 months ago.
ridhim
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ridhim said:

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

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  • Posted 2 months ago.
monali tanna
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monali tanna said:

it was nice attempt but needs 2 hav more precise explanation

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  • Posted 2 months ago.
oLahav
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oLahav said:

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

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  • Posted 2 months ago.
rohitv211
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rohitv211 said:

nice lesson…..easy to understand. : )

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  • Posted about 1 month ago.
akshay kalambur
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akshay kalambur said:

real good and this one is very good

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  • Posted 3 days ago.
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