Friday Quiz, Take I

To keep people sharp on the last day of the week, I'm launching a Friday quiz.

Each week, I'll offer a few problems from my class. This is a chance for some of you to see what you remember from your high school algebra.

I also want to mix it up with more challenging questions, and so I need your help. Please send any submissions for the Friday quiz to my email at chandlerm@washpost.com, and I will choose some to post as we go along for extra credit.

I'll kick things off with some factoring problems we did this morning.

Factor completely!

6x*2 + 27x -15

27x*3 - 64

2d*3 - 10d*2 + 3d -15

3x*2 -12

Have fun! By next week, I will try to find a workable application so I can present exponents and fractions, etc. properly on the blog. For now *3 means to the third power.


By Michael Alison Chandler  |  November 21, 2008; 9:40 AM ET  | Category:  Friday Quiz
Previous: Financial Literacy For Foster Kids (and for me too) | Next: Math Standards Redux

Comments



This might help with the notation: http://www.sitmo.com/latex/

You can type in what you want, then copy and paste it to your page (at least it works on wikis!).

As an Algebra 2 teacher, I enjoy reading your take on things. Thanks for the insight!

Kristen

Posted by: kristenfouss | November 21, 2008 12:39 PM | Report abuse

Michael, for math notation, it's common to use the carat, ^ (Shift-6), for exponents. I've also seen ** to mean exponent, I believe it's used in the Fortran programming language. But I'd use ^ for now.

As for the factoring, it's been a while. Here are my answers:

1) 6x^2+27x-15 = (6x-3)(x+5)
2) 27x^3-64 = (x-4/3)(27x^2+36x+48)
3) 2d^3-10d^2+3d-15 = (d-5)(2d^2+3)
4) 3x^2-12 = 3(x-2)(x+2)

Without showing my work, I'll just say that the first and last are pretty straightforward, but for #2 and #3, I had to use some long division and a lucky guess!

Also, while we're doing math problems, allow me to return to my cow grazing. It's tough to solve without drawing a diagram, but Dr. Math does a very good job of explaining the solution, complete with ASCII artwork. http://mathforum.org/library/drmath/view/53763.html If you're not up to speed on calculus, there's even a link there to a solution that approaches it geometrically. Dr. Math is a good resource, I've entertained myself browsing some of their problems.

Posted by: tomsing | November 21, 2008 1:43 PM | Report abuse

> 2) 27x^3-64 = (x-4/3)(27x^2+36x+48)

well done, tomsing. but check this out:
there's a common factor (of 3)
in the trinomial ... "factor it out":
(x- 4/3)3(9x^2+12x+16)
and then "factor it in" (okay, "multiply it")
on the binomial:
(3x-4)(9x^2+12x+16).

this is to be preferred because
it's now got integer coefficients.

all made transparent if one merely knows
the "difference of cubes" formula:
a^3 - b^3 = (a-b)(a^2 + ab + b^2).
(in our example,
27x^3 - 64 = (3x)^3 - (4)^3
etcetera (i.e., put a = 3x and b =4
and turn the crank).

i had reason to remark to a class this week
that (a^n - b^n) *always* has a factor
of (a-b) ... since you know polynomial
long division, you can find out the
*other* factor easily enough ...

http://vlorbik.wordpress.com

Posted by: vlorbik | November 21, 2008 4:57 PM | Report abuse

Hey- I figure out how to make superscript in here with unicode, at least in Firefox. Just use the entity codes for first power through cube. Anything above three has a code only, like #8308 for 4th power. (see link below)

6x² + 27x -15

http://tlt.its.psu.edu/suggestions/international/bylanguage/mathchart.html#super

Posted by: staticvars | November 21, 2008 5:19 PM | Report abuse

If you want to keep your factoring skills sharp, join yahoo! (have a userid there) and get into yahoo! answers. Lots of opportunities to look at factoring problems there... (kids, younger and older, asking for help with their math homework!)
It's a quick low-effort way to test your chops. Your answer may be wrong and five other contributors may be right, in which case you've learned something.

Posted by: KathyWi | November 22, 2008 12:50 PM | Report abuse

Nice call, vlorbik. I wasn't aware of the a^3-b^3 rule, or maybe I just forgot it.

As for Yahoo Answers, I recently started using it...actually, it was when I was talking about the cow problem, Googled it, and found someone else had asked the question on YA. Since the question had closed, the only way to respond was to leave a comment, which you can't do until you've earned some points. So I started answering questions to earn the points.

It's kind of interesting, but you get the sense that there are a LOT of kids who just want you to do their homework. I stay away from the ones with a list of 5 or 10 problems, and I try to answer in a way that helps them learn something.

Posted by: tomsing | November 22, 2008 9:12 PM | Report abuse

tomsing:

I have a personal connection to this formula as I came up with it (of course, it was well known just not to me) when I was in highschool and used it often.

It factors like this always:

a^n - b^n = (a-b)(a^n + a^(n-1)*b + a^(n-2)*b ... + b^n

So this is a nice pattern. At the same time I also came up with another,

a^n + b^n = (a+b)(a^n - a^(n-1)*b + a^(n-2)*b ... + b^n)

if n is odd.

a^n - b^n = (a+b)(a^n - a^(n-1)*b + a^(n-2)*b ... - b^n)

if n is even.

Posted by: mathlete | November 25, 2008 2:38 PM | Report abuse

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