# How to factor

Does the sight of a number or expression accompanied by the instructions, "Factor completely," strike fear into your heart? Wish you paid attention in algebra? Well, this instructable will teach you how to factor any number, or eligible expression such as Ax^2 + Bx + C.

## Step 1: Factoring numbers

First off, what is a factor?

"Natural number factors" are the complete set of whole numbers, where if you multiply one number in the set by another in the set, you get the number that you're factoring.

For example, the number 5 has two factors: 1, and 5. The number 6 has four factors: 1, 2, 3, and 6.

"Integer factors" include negative numbers.

The number 5 in this case would have four factors: -5, -1, 1, and 5. 6 would have eight factors: -6, -3, -2, -1, 1, 2, 3, and 6.

(Natural numbers are numbers without fractions, starting from 1, 2, 3, 4, 5... all the way up to infinity. Integers are natural numbers, as well as their negative counterparts and 0, or ...-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...)

Factoring numbers with the natural number set is simple. Every number has at least two factors. To find other factors, start dividing the number starting from two and working your way up until you reach that number divided by 2. Any quotient that does not have a remainder means that both the divisor and the quotient are factors of that number.

Say you need to factor the number 9. You can't divide by two evenly, so we skip it. (Note the solution, 4.5, so you know when to stop later on.) 9 is divisible by 3, so add 3 to your list of factors. Work your way up until you divide by 5 (9 divided by 2, rounded up). You'll end up with 1, 3, and 9 as a list of factors.

When factoring numbers in the integer set, you can just add the negative equivalent of your solutions from natural number factoring in. So 9 would have factors of -9, -3, -1, 1, 3, and 9.

Factoring negative numbers can only be done with integer factoring. The solution is the same one you get factoring the positive version of the number. -9 has factors of -9, -3, -1, 1, 3, and 9.

Zero is the only integer that has an infinite amount of factors, and is the only one that has zero as a factor.
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bobx4 months ago

Hello,

How to factor; 1-x^9 - 7 and also how to factor; 2-x^8 - 7 ?

Thanks,

jdevlin826 months ago

x^2 - 4 is not a perfect square, it's a difference of squares. No pun intended, but there's a difference. Take note of the fact that it gets factored out to (x + 2)(x - 2).

Now x^2 - 8x + 16 is a perfect square.
It gets factored out to (x - 4)(x - 4), or just (x - 4)^2. That's what makes it a perfect square. That is not the same as x^2 - 4, which as I mentioned is a difference of squares, because x^2 and 4 are both perfect squares and we are subtracting one from the other.

A perfect square is basically a binomial expression that is a monomial multiplied by itself (squared). It will always result in a trinomial. Even the simplest monomial squared will result in a trinomial. Consider the monomial (x + 1). We see that:

(x + 1)^2 = (x + 1)(x + 1) = x^2 + 2x + 1

Or the negative:

(x - 1)^2 = (x - 1)(x - 1) = x^2 - 2x + 1

6 months ago

Edit: Several times I used the term 'monomial' where I meant 'binomial'.

alatham12 years ago
When you plug your example into the quadratic equation variable 'c' becomes negative for some reason that I cannot see.

(-3 +/- sqrt (3^2 - 4(1)(2)))/2(1)

If I am incorrect I would appreciate feedback as to where I am in error.
urg2 years ago
i really dont get it !
sandyr19503 years ago
These things that are being factored are not EQUATIONS as stated in the text because they do not contain equal signs. They are called EXPRESSIONS. This is important because different things can be done with equations (such as solving) than can be done with expressions.
Phoenixsong (author)  sandyr19503 years ago
Will fix, thanks.
!!!Jenna!!!3 years ago
I don't understand how to simplify radical expressions. The whole Factor, Seperate, Simplify thing confuses me. I get most of it except how they factor and where they get all the random numbers from. I guess I just don't understand how to factor. For example, one of the questions was :...........

3(cube root) and then the radical sign and then 24n^2 x 3(cube root) and then the radical sign and then 36n^2..............

sorry i don't know how to make it look like the actual problem! How would I factor and solve that? Because obviously i am too dumb to figure it out on my own. online school is hard for me.
Phoenixsong (author)  !!!Jenna!!!3 years ago
Also, as your question isn't directly related to the instructable, you'd be better off asking it on Yahoo Answers or similar.
Phoenixsong (author)  !!!Jenna!!!3 years ago
By "and then" did you mean the two terms were set equal to each other? If so, cube both sides so you're left with:
24n^2 = 36n^2
And then move the 24n^2 to the other side of the equation:
36n^2 - 24n^2
Factor out the n^2:
n^2(36 - 24) = 0
Solve.
n = 0

If you meant that the two terms were added together, that's unfactorable.
bseibert3 years ago
Kumon's Math Study Guide provides this question:

Know how to "factor" Binomials:
Problem: -3x^2 + 7x

I thought the obvious answer is pulling out the x for the original binominal which would result in x (-3x+7)

Although I haven't seen this before, I could take each of the factors and have them equal 0 to come up with the answer. Am I missing something?

Example: x = 0 and -3x + 7 = 0
Solve for x, x = 0 and x = 7/3
Phoenixsong (author)  bseibert3 years ago
You are exactly right. The problem should have been worded differently (something like 'Given -3x^2 + 7x = 0, solve for x.').
popewill3 years ago
Oh no. I hate factoring and FOILing with a passion. I thought it was the weekend but i guess you can never escape math :/.
mogg4 years ago
(2x^2 +5x^7) (1+3x) ->
x^2(2+5x^5)(1+3x)

Phoenixsong (author)  mogg4 years ago
Nice catch!
neomancer454 years ago
This instructable and all possible future ones would be easier to read if you used the conventional sign indicating a power, the carrot. (^)

2x2 is strange looking, and could b thought of as 2 times x times 2

also, this gets more complicated when u have variable powers, as x to the power of y, which is different when written as xy.

ex:
x to the power of 4
wrong:
x4
right:
x^4
Phoenixsong (author)  neomancer454 years ago
When this article was written, instructables turned anything between two carets into superscript. It looks like that functionality is not working properly at the moment; although it looks fine in the editor, the formatting does not apply in the published version.
Phoenixsong (author)  Phoenixsong4 years ago
As this has been an issue for a while, I'm assuming instructables has gotten rid of the superscript feature, and I'll insert carets where necessary. Thanks for reminding me.
ilovetine4 years ago
can you please give an example of factoring binomials ?
Mattonator6 years ago
In the binomials section he tries to explain how to factorise 2x2-4x, to get to the answer 2x(x-2). He did this by expanding the binomial function. The common factor in the equation 2x2-4x is 2x, this is because there is a 2x in 2x2 obviously but a 2x inside -4x also. therefore if you times 2x(x-2) out you will get 2x2-4x. This is because 2x times x = 2x2 which is the first part of the equation and 2x times -2 gives the answer -4x because a positive times a negative is a negative number.

Hope this explains it, if not, reply to me
5 years ago
factor, not factorise. you just failed at trying to look smart.
5 years ago
Need more explanation, please. For example, how would you factor 4v squared - 81t squared. I cannot figure it out from the instructable above, or yours. HELP!
4v2 - 81t2

Square root both of the terms in the function. You'll end up with 2v and 9t.

Then, plug it in to the set pattern, (x + y)(x - y).

(2v + 9t)(2v - 9t)
5 years ago
Thanks! I did some research and found the same answer you did. I just wasn't quite understanding it before - thanks again!
Mister nobodie6 years ago
Rated 0.5 (worthless). And thats really what this is. You didn't even explain how you got to 2x (x - 2) for the binomials section. This is pretty much what you did: HOW TO FACTOR 1) factor problem 2) get answer.
5 years ago
Nerd rage.
bassclarinet235 years ago
My teacher gave us a song to memorize the Quadratic Equation...

(Imagine the tune of "Pop! Goes the Weasel".)

The opposite of lower case "B",
Plus or minus the square root,
Of "B" squared minus 4(ac),
All over 2a.

Fin.

It's irritating, but you'll remember the formula.

Pronounce...
4(ac) = four "A" "C"
2a = two "A"

Phoenixsong (author)  bassclarinet235 years ago
I was taught it to the tune of Row, Row, Row your boat:

x equals minus b,
plus or minus root,
b squared minus four a c,
all over two a.

:)
5 years ago
Ha ha. Cool.
mi9d6 years ago
Sorry, but shouldn't n=3.x.x ?
I mean 9.x.x times 9.x.x = 81.x.x.x.x

( sorry, can't type the power superscripts )
6 years ago
9x2 times 9x2 is 81 * 81 which = 6561 which is the same as 9*9*9*9 or 94.

814 is 43,046,721
Phoenixsong (author)  mi9d6 years ago
Oh, and btw, to type power superscripts, put a carrot (shift+6) both in front of and in back of the number/word/phrase that you want in superscripts.
Phoenixsong (author)  mi9d6 years ago
Gotcha. Thanks for catching that.
Shinta7866 years ago
oh man, I just scrolled to the comments in 5secs. don't dare to read this, but must be good
I hate factoring lol - we have to do this huge 125 Q. packet for Algebra II. If I need help, I'll be sure to reference this. ;D
Blazo6 years ago
This is a good instructable and i dont know about you but we learned this in 7th grade. By the way im from Macedonia and at 7th grade kids are usualy 12-13 yrs old. :D
menace6 years ago
ax4+bx3+cx2+dx+e
synthetic division
normal division those would be reallly helpful.
Phoenixsong (author)  menace6 years ago
If you wanted to use long division instead of synthetic division, you could divide the polynomial through by x - P/Q. (If that's what you mean...) Synthetic division goes by quicker, though.
josefu06 years ago
omg this schooling is boring.
Phoenixsong (author)  josefu06 years ago