Factoring PolynomialsDate: 11/07/2001 at 20:28:37 From: Emily Subject: Factoring I'm having trouble factoring things like: 12x^3y^9 + 20x^5y^4 ^ = exponent and I'm also having trouble FOILING. Please help - I need it explained better. Date: 11/08/2001 at 14:51:38 From: Doctor Ian Subject: Re: Factoring Hi Emily, Let's take a look at 12x^3y^9 + 20x^5y^4 If we break the constants into prime factors, and expand out the exponents, we have 2*2*3*x*x*x*y*y*y*y*y*y*y*y*y + 2*2*5*x*x*x*x*x*y*y*y*y Yuck! But now we can identify the pieces that the terms have in common. 2*2*3*x*x*x*y*y*y*y*y*y*y*y*y + 2*2*5*x*x*x*x*x*y*y*y*y --- --- 2*2*3*x*x*x*y*y*y*y*y*y*y*y*y + 2*2*5*x*x*x*x*x*y*y*y*y --- ----- --- ----- 2*2*3*x*x*x*y*y*y*y*y*y*y*y*y + 2*2*5*x*x*x*x*x*y*y*y*y --- ----- ------- --- ----- ------- And then we can use the distributive property to move the shared parts out front: 2*2*x*x*x*y*y*y*y(3*y*y*y*y*y + 5*x*x*) Converting back to exponents, this gives us 4x^3y^4(3y^5 + 5x^2) Now, this is kind of messy, so we could do the same thing while retaining the exponents: 12x^3y^9 + 20x^5y^4 x(12x^2y^9 + 20x^4y^4) Factor out an x. x^2(12xy^9 + 20x^3y^4) Again. x^3(12y^9 + 20x^2y^4) Again. x^3y(12y^8 + 20x^2y^3) Factor out a y. x^3y^2(12y^7 + 20x^2y^2) Again. x^3y^3(12y^6 + 20x^2y) Again. x^3y^4(12y^5 + 20x^2) Again. 2x^3y^4( 6y^5 + 10x^2) Factor out a 2. 4x^3y^4( 3y^5 + 5x^2) Again. When you get comfortable doing it the long way, you can start using shortcuts. For example, in this case, we can look at 12x^3y^9 + 20x^5y^4 and see that the exponent of x is at least 3 in each term, so we can factor out x^3, subtracting 3 from each exponent: x^3(12y^9 + 20x^2y^4) imilarly, we can see that the exponent of y is at least 4 in each term, so we can factor out y^4, subtracting 4 from each exponent: x^3y^4(12y^5 + 20x^2) The greatest common factor of 12 and 20 is 4, so we can factor that out, dividing each coefficient by 4: 4x^3y^4( 3y^5 + 5x^2) So this is quicker! But if you try to follow 'factoring rules' without understanding what's really going on - i.e., that you're really just applying the distributive property over and over until there is nothing left to distribute - then you're likely to get mixed up and make a mistake. It's better to do things the long way until they get so easy that you find yourself looking for a shorter way. As for FOILing, don't bother. Use the distributive property instead. It makes more sense, and is useful in more situations: Distributive Property: (x+2)(x+4) http://mathforum.org/dr.math/problems/lydia.09.18.01.html When FOIL Fails http://mathforum.org/dr.math/problems/ryan.03.22.01.html Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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