Why Factor?Date: 08/18/2001 at 16:54:19 From: Kevin O'Neil Subject: Factoring in algebra I haven't seen this answer yet anywhere I've looked. WHY specifically does one have to factor out a problem? My daughter is homeschooling and we have an excellent Algebra 1 teacher, but it seems to be a hard question to answer. I've been explained the how's, the where's, the what's, but not the WHY. My daughter seems to need a practical explanation in order to understand math or how it applies to something. If you had a problem like 4b^2y^2 - 8b^2y + 24b^2 = ? What is the practical use of factoring this out? I know it's like unmultiplying, but trying to explain that without a reason "why" behind it gets my daughter confused. She wants to know where it is used and why it is used, to help her figure out how. What applications would it serve? An example where a formula like that might be used would be helpful. Why not just solve the problem anyway? Aren't all the individual pieces needed there to solve it? Date: 08/20/2001 at 18:01:07 From: Doctor Ian Subject: Re: Factoring in algebra Hi Kevin, If I understand you correctly, you want to know why the expression 4b^2(y^2 - 5y + 6) = ? is preferable to the expression 4b^2y^2 - 20b^2y + 24b^2 = ? Well, the main reason for identifying common factors is to let you see more deeply into the pattern. In this case, once we get the 4b^2 out in front, we can see that we have a standard quadratic form in y, which we can simplify even more: 4b^2y^2 - 20b^2y + 24b^2 = 4b^2(y^2 - 5y + 6) = 4b^2(y - 2)(y - 3) Now, why is _this_ form preferable? Well, for one thing, if the '?' is a zero, as it is in 'standard' form, then we know that 4b^2(y - 2)(y - 3) = 0 Just by looking at this, we can see that if b is non-zero, there are only two possible values of y that can make this equation true: y = 2 and y = 3. It's also easy to see that halfway in between those two values, 4b^2(2.5 - 2)(2.5 - 3) = 4b^2(0.5)(-0.5) the value of the expression is negative, which means that since we know that every quadratic expression is the description of a parabola, we now have enough information to draw a pretty good graph of the expression. Try doing that with the original form of the equation. Note that this also tells us that if we vary the value of b, we can alter the shape of the parabola, but we can't move the axis of symmetry (since that would require us to change where it hits the x-axis). In general, when mathematicians develop a technique like factoring out common terms, you can be sure it's because they've found that it turns some really tedious and error-prone task (like trying to draw the graph of a function) into a very easy one, e.g., Why study Prime and Composite Numbers? http://mathforum.org/library/drmath/view/57182.html Have you heard the joke in our archives toward the end of this answer? Quadratic Formula: Solve for b http://mathforum.org/library/drmath/view/52370.html Such jokes expose a deep truth about the way mathematicians do mathematics, and I mention them because one way to make learning math more entertaining is to try to approach the subject the way a mathematician would. I'm not saying that you should learn to enjoy jotting down pages and pages of equations! What I mean is that you should get into the habit of looking for the easiest possible way to get from the statement of a problem to its solution, which in many cases means finding a short connection between the problem you're working on and one that you know has already been solved... whether or not you were the one who solved it. Factoring things out is, ultimately, a technique for helping you do this. As you've noted, you can solve problems without factoring out common terms... and you can get from New York to Los Angeles on foot, if you want to, but it will take longer. I hope this helps. Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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