Step 3: 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.


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].
<p>Hello,</p><p>How to factor; 1-<strong>x^9 - 7 and also how to factor; 2-</strong><strong>x^8 - 7 ?</strong></p><p><strong> Thanks,</strong></p>
<p><strong>x^2 - 4</strong> is not a perfect square, it's a <em>difference of squares. </em>No pun intended, but there's a difference. Take note of the fact that it gets factored out to <strong>(x + 2)(x - 2).</strong><br><br>Now <strong>x^2 - 8x + 16 </strong>is a <em>perfect square.<br> </em>It gets factored out to <strong>(x - 4)(x - 4)</strong>, or just <strong>(x - 4)^2. </strong>That's what makes it a perfect square. That is not the same as <strong>x^2 - 4, </strong>which as I mentioned is a <em>difference of squares, because x^2 and 4 are both perfect squares and we are subtracting one from the other.</em><br><br>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 <strong>(x + 1)</strong>. We see that:<br><br><strong>(x + 1)^2 = (x + 1)(x + 1) = x^2 + 2x + 1<br><br></strong>Or the negative:<br><br><strong>(x - 1)^2 = </strong><strong>(x - 1)(x - 1) = x^2 - 2x + 1</strong></p>
<p>Edit: Several times I used the term 'monomial' where I meant 'binomial'.</p>
When you plug your example into the quadratic equation variable 'c' becomes negative for some reason that I cannot see. <br> <br>(-3 +/- sqrt (3^2 - 4(1)(2)))/2(1) <br> <br>If I am incorrect I would appreciate feedback as to where I am in error.
Thank you for sharing this information on <a href="http://www.jdfactors.com" rel="nofollow">factoring</a> binomials. I used to know how to do this, but I forgot it since I've been out of school.
Thanks for sharing. I would love to learn more about <a href="http://www.jdfactors.com" rel="nofollow">factoring</a>. You really break down all the steps. Great job.
Thanks for sharing this! I have been trying to learn to be better at <a href="http://www.jdfactors.com" rel="nofollow">factoring</a> and I really think this will help me a lot! Can you tell me where I can maybe find some more practice problems? Thanks again for sharing!
Thanks for sharing this about <a href="http://www.jdfactors.com" rel="nofollow">factoring</a>. I appreciate it!
I was never very good at math. So when my daughter asked me to help her with <a href="http://www.jdfactors.com" rel="nofollow">factoring</a> homework, I couldn't remember where to even begin. So thank you so much for posting! This has helped us both out a ton!
Thanks so much for sharing this information! I had no clue there were so many ways to use <a href="http://www.jdfactors.com" rel="nofollow">factoring</a> or things to do with that! Can you tell me where I can find more uses of it or how to factor different things! Thanks for sharing!
i really dont get it ! <br>i want to learn more about it <br>
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.
Will fix, thanks.
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 :........... <br> <br>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.............. <br> <br>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.
Also, as your question isn't directly related to the instructable, you'd be better off asking it on Yahoo Answers or similar.
By &quot;and then&quot; did you mean the two terms were set equal to each other? If so, cube both sides so you're left with:<br>24n^2 = 36n^2<br>And then move the 24n^2 to the other side of the equation:<br>36n^2 - 24n^2<br>Factor out the n^2:<br>n^2(36 - 24) = 0<br>Solve.<br>n = 0<br><br>If you meant that the two terms were added together, that's unfactorable.
Kumon's Math Study Guide provides this question: <br> <br>Know how to &quot;factor&quot; Binomials: <br> Problem: -3x^2 + 7x <br> Answer: 0, 7/3 <br> <br>I thought the obvious answer is pulling out the x for the original binominal which would result in x (-3x+7) <br> <br>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? <br> <br>Example: x = 0 and -3x + 7 = 0 <br>Solve for x, x = 0 and x = 7/3
You are exactly right. The problem should have been worded differently (something like 'Given -3x^2 + 7x = 0, solve for x.').
Oh no. I hate factoring and FOILing with a passion. I thought it was the weekend but i guess you can never escape math :/.
(2x^2 +5x^7) (1+3x) -&gt;<br>x^2(2+5x^5)(1+3x)<br><br>Answer has 3 factors (^^)
Nice catch!
This instructable and all possible future ones would be easier to read if you used the conventional sign indicating a power, the carrot. (^)<br><br>2x2 is strange looking, and could b thought of as 2 times x times 2<br><br>also, this gets more complicated when u have variable powers, as x to the power of y, which is different when written as xy.<br><br>ex:<br>x to the power of 4<br>wrong:<br>x4<br>right:<br>x^4
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.
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.
can you please give an example of factoring binomials ?
In the binomials section he tries to explain how to factorise 2x<sup>2</sup>-4x, to get to the answer 2x(x-2). He did this by expanding the binomial function. The common factor in the equation 2x<sup>2</sup>-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 2x<sup>2</sup>-4x. This is because 2x times x = 2x<sup>2</sup> 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.<br/><br/>Hope this explains it, if not, reply to me<br/>
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!
4v<sup>2</sup> - 81t<sup>2</sup><br/><br/>Square root both of the terms in the function. You'll end up with 2v and 9t.<br/><br/>Then, plug it in to the set pattern, (x + y)(x - y).<br/><br/>(2v + 9t)(2v - 9t)<br/>
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...<br /> <br /> (Imagine the tune of &quot;Pop! Goes the Weasel&quot;.)<br /> <br /> The opposite of lower case &quot;B&quot;,<br /> Plus or minus the square root,<br /> Of &quot;B&quot;&nbsp;squared minus 4(ac),<br /> All over 2a.<br /> <br /> Fin.<br /> <br /> It's irritating, but you'll remember the formula.<br /> <br /> <br /> Pronounce...<br /> 4(ac) = four &quot;A&quot; &quot;C&quot;<br /> 2a = two &quot;A&quot;<br /> <br />
I was taught it to the tune of Row, Row, Row your boat:<br /> <br /> x equals minus b,<br /> plus or minus root,<br /> b squared minus four a c,<br /> all over two a.<br /> <br /> :)<br />
Ha ha. Cool.<br />
Sorry, but shouldn't n=3.x.x ? <br/>I mean 9.x.x times 9.x.x = 81.x.x.x.x <br/><br/>( sorry, can't type the power superscripts )<br/>
9x<sup>2</sup> times 9x<sup>2</sup> is 81 * 81 which = 6561 which is the same as 9*9*9*9 or 9<sup>4</sup>.<br/><br/>81<sup>4</sup> is 43,046,721<br/>
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.
Gotcha. Thanks for catching that.
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
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
How about factoring polynomials?<br/>ax<sup>4+bx</sup>3+cx<sup>2+dx+e</sup><br/>synthetic division<br/>normal division those would be reallly helpful.<br/>
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.
omg this schooling is boring.
No one said you had to read it. >_>
FOIL ftw! brings back memories of my math tutoring days.
Added section on factoring programs. Thanks.
this is a really nice, deep and informed feature... except in that you use the word 'factor' four times in explaining what a factor is. So unless you already know what a factor is the explanation isn't going to make sense. 1st rule of explanation: assume your student knows nothing. Otherwise, math tutorials are a Damned Fine Plan, please do keep 'em coming :-)

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