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Picture of 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.
 
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Step 1: Factoring numbers

Picture of 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.

Step 2: Factoring the GCF from an expression

Picture of Factoring the GCF from an expression
And no, I don't mean factoring the expression of your boss as you tell him you accidentally flooded the break room with coffee.

Algebraic expressions consist of numbers, which are called coefficients, and variables, which can be raised to a power. In the expression x^2 + 6x + 8, 1 is the coefficient of x^2, the variable. (If you do not see a coefficient before a variable, it's a 1, because x^2 is multiplied by 1.) Likewise, 6 is a coefficient of x^1. (A lone variable is raised to a power of one.) 8 is called a constant - it is not multiplied by a variable. (You can visualize it being multiplied by x^0, and any number raised to the 0th power is equal to 1).

To factor an expression, you have to start by factoring out the GCF, or Greatest Common Factor. List the factors of each component of the expression. Here we are interested in finding the natural number factors.

The expression x^2 + 6x + 8 would have factors that look like this:

x^2: 1
6x: 1, 2, 3, 6
8: 1, 2, 4, 8

If you look at the three lists, there is only one thing that they all share in common, the number one. This means there is no coefficient greater than one to factor out.

Then you look at the exponents' powers. 2, 1, and 0. If you see a zero, the expression cannot be factored by a variable.

This expression is ready for the next step.

Here is an example that does have a GCF that needs to be factored out: 2x^3 + 18x^2 + 10x. Factor each part:

2x^3: 1, 2
18x^2: 1, 2, 3, 6, 9, 18
10x: 1, 2, 5, 10

Here we can see that the parts have 1 and 2 in common. We find the largest number, 2.

Then we look at the powers of exponents: 3, 2, and 1. Find the smallest number that isn't 0, in this case the number one. That means x^1, or simply x, can be divided into the expression.

Multiply the number and variable together to get 2x. Then divide each part of the expression by 2x.

2x^3 / 2x = x^2
18x^2 / 2x = 9x
10x / 2x = 5

The expression with the GCF factored out is 2x (x^2 + 9x + 5). Note that you must put the factored expression in parentheses and write the GCF next to it.

Step 3: Factoring binomials

Picture of Factoring binomials
Binomials are expressions with only two terms being added.

2x^2 - 4x is an example of a binomial. (You can say that a negative 4x is being added to 2x2.)

First, factor out the GCF, 2x. You're left with 2x (x - 2). This is as far as this binomial can go. Any binomial in the form 1x +/- n cannot be factored further.



When you have a binomial that is a variable with an even exponent, added to a negative number that has a square root that is a natural number, it's called a perfect square.

x^2 - 4 is an example of this. It can be expressed as the product of the square root of the variable plus the square root of the positive constant, and the square root of the variable minus the square root of the positive constant.

Huh?

Basically, take the square root of the variable. You'll end up with x. Then square root the 4. You'll end up with 2. If you add them together, you'll get x+2. Subtract them, and you'll get x-2. Multiply the two, and you'll get (x+4)(x-4). You've just factored a perfect square.

If you multiply (x+2)(x-2) together using FOIL, you'll end back up with x^2-4.

(FOIL: First Outer Inner Last, a way of multiplying two binomials together. Multiply the first terms of the binomials (x and x in this case), then the outer two (x and -2), then the inner two (2 and x), then the last terms (2 and -2), then add them all up. x^2 - 2x + 2x - 4 = x^2 - 4.)

This can be done again if one of the binomials is a perfect square, as in this instance:

x^4 - 16 = (x^2 + 4) (x^2 - 4) = (x^2 + 4) (x + 2) (x - 2).

This can be factored further if you bring in irrational numbers, see step [9].



How to factor binomials in the form of (x^3 + b^3):

Just plug into (a - b) (a^2 +ab + b^2). For example, (x^3 + 8) = (x - 2) (x^2 + 2x + 4).

How to factor binomials in the form of (x^3 - b^3):

Plug into (a + b) (a^2 - ab + b2). Note that the first two signs in the expression are switched.

(x^3 - 8) = (x + 2) (x^2 - 2x + 4).

Both examples can be factored further once you learn how to factor trinomials in step [4].

Step 4: Factoring trinomials

Picture of Factoring trinomials
step4-2.bmp
Trinomials: An expression with three terms added together. 2x^2 + 6x - 8 will serve as our lucky demonstrator.

First, factor out the GCF. This will ALWAYS be your first step when factoring ANY expression.

2 (x^2 + 3x - 4)

If you end up with a power of x greater than two after factoring out the GCF, move on to another step.

List the integer factors of the constant. You'll want two pair them up like so:

-4, 1
-2, 2
-1, 4

You want to find one of these that when added up equals the coefficient of the second term, 3. -1 + 4 = 3. From here, write out two sets of parentheses with x's inside:

(x ) (x )

Then stick the two terms that worked into the parentheses.

(x - 1) (x + 4)

Don't forget to add the GCF back.

2 (x - 1) (x + 4)

That's how you factor a trinomial.

Here's another one: 2x^2 + 11x - 6.

There's a twist this time: The coefficient of x^2 is not 1. This means that we will be adding another step:

List factors of the constant, -6, as well as the coefficient of x2, 2.

-6, 1
-3, 2
-2, 3
-1, 6

1, 2

Now, you'll want to multiply each of the factors on the left side by 1, and on the right by 2. Repeat by switching the 1 and 2. You'll end up with

-6, 2
-3, 4
-2, 6
-1, 12
-12, 1
-6, 2
-4, 3
-2, 6

Find the pair that adds up to the middle term's coefficient, in this case, -1 + 12 = 11. Set up the parentheses:

( x ) ( x )

Stick in the original numbers (that you had before multiplying by 1 and 2):

( x - 1) ( x + 6)

Then stick in the one and two as coefficients of x so that when you multiply the outer and inner terms and add them together, you'll get 11.

(2x - 1) (x + 6)

If you check your work by FOILing it out, you'll end up with 2x^2 + 11x - 6, the expression you started with. Congrats!

Step 5: Factoring trinomials by substitution

Picture of Factoring trinomials by substitution
9x^4 + 45x^2 + 14.

Don't you think this expression would be easier to factor with smaller numbers and variable powers?

You can substitute a lower number and variable power like so:

Set n = 3x^2 (the GCF of the variable powers, and the square root of the GCF of the coefficients of numbers multiplied by a power of x). Then substitute it in by dividing the terms in the original expression by n.

n^2 + 15n + 14.

Now you can easily factor.

(n + 14) (n + 1).

Stick the 3x^2 back into the expression where the n's are.

(3x^2 + 14) (3x^2 + 1).

Step 6: The quadratic equation

Picture of The quadratic equation
solution2.PNG
If none of the combinations you get (from step 4) add up right, you'll have to use the quadratic equation.

(-b +/- sqrt (b^2 - 4ac))/2a

(sqrt (#) = square root of #)

Where a trinomial has the form ax^2 + bx + c.

So, if you wanted to use the quadratic formula with 1x^2 + 3x + 2, you'd plug in like so:

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

This simplifies down to (-3 +/- sqrt 17)/2. The factors of 1x^2 + 3x + 2 would be (x - ((-3 + sqrt 17)/2)) (x - ((-3 - sqrt 17)/2)). (You stick the answer to the right of an "x - ". More on why that works, in step [8].)

Step 7: Factoring polynomials by grouping

Picture of Factoring polynomials by grouping
Sometimes you will get four or more terms, that look something like this:

2x^2 + 6x^3 + 5x^7 + 15x^8

There is no common coefficient, and factoring out x^2 doesn't help much. This is where you would use grouping to factor.

Grouping means factoring out the GCF of only two terms of the expression. You can see that 2x^2 + 6x^3 and 5x^7 + 15x^8 both can have a GCF taken out. Do so.

2x^2 (1 + 3x) + 5x^7 (1 + 3x)

Note that there is a common factor, 1+3x. This expression can be rephrased to (2x^2 + 5x^7) (1 + 3x). There's your answer.

Note that (2x^2 + 5x^7) (1 + 3x) can be factored further by factoring out an x^2 from the first binomial: x^2 (2 + 5x^5) (1 + 3x).

Step 8: Factoring polynomials by synthetic division

Picture of Factoring polynomials by synthetic division
solution.PNG
Sometimes you'll get beastly polynomials that look like they have no hope.

3x^3 + 8x^2 - 9x + 2 is an example. You can't use grouping to factor out a GCF in a way that would produce a common factor.

In order to explain how this works, you need to know that when solving an equation by factoring, you need to set the factored out thing equal to 0 and find out what X equals so that it equals zero. For example, 0 = (x - 2) (x + 1). The solutions are 2 and -1.

If a polynomial has integer coefficients, every zero, or solution, has the form P/Q, where P = a factor of the constant term, and Q = a factor of the leading coefficient.

Basically, if you list all the factors of the constant, and divide them by the factors of the leading coefficient (the coefficient next to the variable with the highest power) in every combination, you will get a list of possible rational solutions. How does this help you factor? If you get 2 as a solution, you can work backwards and say that one of the factors of the equation was (x - 2).

So, back to the example:

Factors of 2: +/- 1, +/- 2 (you need to include negatives)
Factors of 3: +/- 1, +/- 3

P/Q: +/- 1, +/- 1/3, +/- 2, +/- 2/3

Once you have your list, you'll use something called synthetic division to see which of those P/Q's are actually solutions.

Synthetic division is a way of dividing polynomials by a binomial of the form x-k. I'm not going to explain how it works, but just show how to use it for factoring.

First, put one of your P/Q's in a little box or set of parentheses, then list the coefficients and constant in a row next to it. If the polynomial skips a power (x^2 + 2) then you need to add a 0 for where x1 should have been.

(Expression: 3x^3 + 8x^2 - 9x + 2)

(Ignore the asterisks, they're used as placeholders. Better yet, see the first picture.)

(1) 3 8 -9 2



Leave a blank space, draw a line, then drop the first term, 3, down.

(1) 3 8 -9 2


***3

Then multiply it by the number in the box and put it beneath the next term.

(1) 3 8 -9 2
******3

***3

Add 8 + 3

(1) 3 8 -9 2
******3

***3 11

Multiply.

(1) 3 8 -9 2
******3 11

***3 11

Add.

(1) 3 8 -9 2
******3 11

***3 11 2

Multiply.

(1) 3 8 -9 2
******3 11 2

***3 11 2

Add.

(1) 3 8 -9 2
******3 11 2

***3 11 2 4

That string of numbers, 3, 11, 2, 4, gives you an expression with one degree less (if the highest exponent in the original expression is 3, the highest exponent in the quotient will be a 2) as well as a remainder.

(Original Expression: 3x^3 + 8x^2 - 9x + 2)

Quotient: 3x^2 + 11x + 2 Remainder 4

If you get a remainder, then the number in the box you tried is not a solution for the equation. Cross that number off your list, and try again with another number. It's pretty much guess and check.

Eventually you'll try 1/3 and you'll find that it divides through cleanly. You'll end up with:

(x - 1/3) (3x^2 + 9x - 6).

Now that you have a trinomial of power two, you can go back and factor it. Don't forget to take out the GCF first! You're left with (x - 1/3) (3) (1x^2 + 3x + 2). Factor out the trinomial via the quadratic equation (this equation was used as an example in step [6], so refer back if you need to). You'll end up with (3) (x - 1/3) (x - ((-3 + sqrt 17)/2)) (x - ((-3 - sqrt 17)/2)). Very ugly, but that's how you do it.

Step 9: Factoring further: irrationals and imaginaries

Picture of Factoring further: irrationals and imaginaries
Binomials number without a perfect root being subtracted from a squared variable like (x^2 - 2) can be factored further using square roots. (x + sqrt (2)) (x - sqrt (2)). This brings in the irrational set of numbers.

Binomials with a number added to a squared variable like (x^2 + 1) can be factored further using imaginary numbers. "i" stands for the square root of negative one. So (x^2 + 1) can be factored into (x + i) (x - i). This brings in the imaginary set of numbers.

Step 10: Huzzah!

Picture of Huzzah!
You now know how to factor any number or expression you'll probably ever come across. Good for you!

There are also programs out there that can do this for you. If you google "polyroot" you'll get links to a few programs for your computer. The HP 39/40gs graphing calculators have the polyroot function built in. If you have a TI-89 graphing calculator, it also has a factoring function. Earlier model TI graphing calculators don't have it built in, but they do have factoring programs. Google "ti quadratic solver" for programs you can transfer to your TI graphing calculator.

You can also find real solutions to quadratic equations by graphing them and using the 'zero' function to calculate where the graph intersects the x-axis. You can then stick that number next to a "x -".

Disclaimer: Most math classes either disallow calculators that can factor, or make you clear the memory (along with programs) of programmable calculators. Also, if any solutions have a non-natural root in them, you'll get a long string of decimals which is unsuitable as an answer. Just learn how to do it by hand.
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bobx3 months ago

Hello,

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

Thanks,

jdevlin825 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

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 !
i want to learn more about 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
Answer: 0, 7/3

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)

Answer has 3 factors (^^)
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 ?
Mattonator5 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
factor, not factorise. you just failed at trying to look smart.
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!
Phoenixsong (author)  ladams2345 years ago
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)
Thanks! I did some research and found the same answer you did. I just wasn't quite understanding it before - thanks again!
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.
Nerd rage.
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.

:)
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 )
Mattonator mi9d5 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
How about factoring polynomials?
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
No one said you had to read it. >_>
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