Do You Dream of Factoring?

Factoring is one of the most important skills that students learn in algebra. My teacher at Fairfax High told the class: "You will practice factoring until you are dreaming of factoring!"

For today's Friday quiz see if you can you answer these questions from a factoring worksheet we did earlier this year.

By Washington Post Editors  |  January 30, 2009; 10:24 AM ET  | Category:  Friday Quiz
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I hope the factoring worksheets they are assigning in school are not multiple choice. Factoring should not be multiple choice!

Posted by: DCMathTutor | January 30, 2009 11:31 AM | Report abuse

I think the last question in the quiz is wrong. The supposedly correct most factored answer had (4x^2-1) but this factors as (2x+1)(2x-1) as in one of the other answers.

Posted by: mkness159 | January 31, 2009 3:02 AM | Report abuse

Whew! I'm glad I still remember how to do that.

Posted by: bokamba | January 31, 2009 8:23 AM | Report abuse

Actually the use of multiple choice factoring questions can help the student to understand the "why" in factoring rather than just be mechanical. In each option, if the student eliminates what cannot be correct, then they have the correct answer ... and eliminating the incorrect options is both very easy and quick, IF AND ONLY IF the student understands the basic principles. Once the final response is left standing, it is easy for the student to verify that it is correct. Teaching the student to think is where we need to go in the 21st century. Rote memorization and trial and error approaches served us well in the 20th century, but being on top in the 21st century will require us to teach our students more than just rote and memorized skills.

Posted by: hradvocate | January 31, 2009 8:55 AM | Report abuse

mkness159 is, of course, right. The correct answer to the last problem (which asked for the COMPLETE factorization) is (2x-1)(2x+1)(x+1). I hope students didn't lose points for giving the "wrong" (but correct!) answer!

Posted by: brian137 | January 31, 2009 9:19 AM | Report abuse

mkness159 is, of course, right. The correct answer to the last problem (which asked for the COMPLETE factorization) is (2x-1)(2x+1)(x+1). I hope students didn't lose points for giving the "wrong" (but correct!) answer!

Posted by: brian137 | January 31, 2009 9:21 AM | Report abuse

I think that hradvocate has hit on the essence of this debate, but I disagree with some of your conclusions. Specifically, the ability and habit of eliminating incorrect options, and checking one's answer, is an excellent skill, far more general and important than factoring. However, in this context eliminating incorrect options is testing one's ability to multiply polynomials which is not exactly the same thing as factoring (factoring polynomials also involves factoring numbers into a product of primes, e.g. 18 = 2*3*3.)

One could even argue that this worksheet would be better titled 'multiplying polynomials' than 'factoring.' The question is essentially which product of polynomials is equivalent to ...

There are certainly cases where it seems easier to verify that an answer is correct then it does to come up with the correct answer on ones own (think of encryption - it sees harder to discover an encryption scheme then it would be to verify that one is correct.) However, in many cases multiple choice exams can require you to do more than verify the correctness of an option. For example, if we want to do more to test factoring skills we could provide less scaffolding by changing the question to ask the student to identify one of the terms in the factor with options
a. (x-27)
b. (x+9)
c. (x+18)
d. (x-3)

Though this concept may not be new, I only recently became aware of it through Jeffery Ullman's idea of 'root questions' that he advocates as part of 'Gradiace On-Line Accelerated Learning'

Posted by: ford36 | January 31, 2009 9:57 AM | Report abuse

mkness159 and brian137 you are right. There are multiple errors in the last problem primarily centered around notation. Pay close attention to the little x and big X. This question has no correct answer as a possible option.

Option B:
(4X^2-1)(X+1) = 4X^3 + 4X^2 - X - 1. This is not 4x^3 + 4X^2 - X - 1

Option C:
(2X-1)(2x+1)(X+1) = (4Xx + 2X - 2x - 1)(X+1) = 4X^2x + 2X^2 + 2Xx + X - 2x - 1 This also is not 4x^3 + 4X^2 - X - 1

Are you pulling my leg Chandler? ;)

Posted by: mcross80 | January 31, 2009 10:59 AM | Report abuse

If the polynomial to factor in problem 4 were 4x^3+4x^2+x+1 or 4x^3-4x^2+x-1, then one of the factors would be 4x^2+1, which can't be factored into linear terms using real, rather than complex, numbers. Then an answer similar to the middle one would be correct. Maybe that's what the testers were trying for ...

ford36 is right that these problems are more about "multiplying polynomials" than "factoring polynomials". A better check of factoring skill is to ask if particular linear or quadratic polynomials divide the given polynomial exactly. Then you can use information such as that the constant term of a given polynomial equals the product of the constant terms of the factors (and similar information regarding other coefficients) to help find the answer.

Posted by: rbtorrance | January 31, 2009 8:01 PM | Report abuse

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