Factoring Non-Monic Quadratics

Regardless of the course, non-monic factoring was always a thorny issue in my teaching.  My students never seemed to get “good” at it, even though they seemed alright at monic factoring (“monic” just means the first coefficient is 1, like x^2 + 14x + 48).  This topic made me really question why I was teaching it, for several reasons:

  • The very next thing was the quadratic formula, and for most of the things non-monic factoring could be useful for, I felt the quadratic formula would be just as good.
  • I couldn’t find many places later in my curriculum where non-monic factoring was being used, so it seemed like a topic taught for a single purpose.
  • The methods I saw for factoring non-monic quadratics had little or nothing to do with the methods for factoring monic quadratics.
  • The method I learned and first taught amounted to trial-and-error.

And maybe you know this method too: to factor 6x^2 + 31x + 35, you write down all the factors of 6, separately write down all the factors of 35, and start making pairs.  Eventually you either find the pair that works, or you run out of pairs:

(x+1)(6x + 35) = 6x^2 + 41x + 35, nope

(x+5)(6x + 7) = 6x^2 + 37x + 35, nope

(x+7)(6x + 5) = 6x^2 + 47x + 35, nope, keep trying…

And I was polite in picking 6 and 35 here, two numbers with only two prime factors each!  I feel this method is a mathematical nightmare.  Keep testing, keep checking.  And don’t give the ones that aren’t factorable, since the only way to know it’s unfactorable is to test all the possibilities, and that’s just mean.

A year or two later, I learned and taught the “key number method”: multiply the coefficients of “a” and “c” (6 \cdot 35 = 210).  Then you break up the middle term (31x) into two pieces whose coefficients multiply to 210:

6x^2 + 31x + 35 = 6x^2 + 10x + 21x + 35

Then “group” in pairs and a miracle occurs:

(6x^2 + 10x) + (21x + 35) = 2x(3x + 5) + 7(3x + 5)

= (3x + 5)(2x + 7)

This worked a lot better for my students, by which I mean they got correct answers faster and with greater accuracy.  But the core of this method is the “miracle” that splitting the 31x in this exact, specific way will do great things.  It works because it works.  (There are better explanations, but my students just memorized what to do.)

One advantage of the key number method is it can be applied to monics, too, visualizing the “sum and product” concept:

x^2 + 14x + 48 = x^2 + 6x + 8x + 48

= x(x+6) + 8(x+6)

= (x+6)(x+8)

But this generally comes after the fact: I didn’t teach students to factor monics in this way.

While working on CME Project, I learned (through Al Cuoco and Jeremy Kahan, a field test teacher) about a “scaling” method that uses monic factoring as the core of non-monic factoring.  It feels a lot more natural, cements monic factoring, and fits tightly with Mathematical Practice #7, “Look for and make use of structure.”  It starts with specific non-monics like this one:

25x^2 + 70x + 48

Try factoring that for a second using either of the methods presented above.  It’s messy!  But, would you believe this is actually a monic quadratic?  It’s just got a different variable: 5x.

25x^2 + 70x + 48 = (5x)^2 + 14(5x) + 48

Now cover your finger over each 5x: it reads finger^2 + 14 \cdot finger + 48.  Doesn’t matter what’s under the finger: it factors!

F^2 + 14F + 48 = (F+6)(F+8)

And you’re done when you lift your finger, remembering that F = 5x.  In teaching, I used capital letters for these substitutions, to remind students that there was more work to be done later.

(5x+6)(5x+8)

How fast was that?  And understandable, too!  The core concept of a replacement of variable (the book calls this “chunking”) plays forward deeply into later topics and courses: when I say I used capital letters for substitutions, I generally was doing that with Precalculus or Calculus students, but the concept can be seen much, much earlier.  By using it frequently, it becomes a tool students actively look to use when they see something complicated.

But I fudged the example: it’s got 25x^2.  How about that original one, 6x^2 + 31x + 35?  It doesn’t have a perfect square term, but … wishful thinking … we can make one by multiplying through by 6, then paying it back later.

6(6x^2 + 31x + 35) = (6x)^2 + 31(6x) + 210

\mathbf{= F^2 + 31F + 210}

\mathbf{= (F+10)(F+21)}

Note that this method includes the step that was part of the “key number method”: the 210 is produced by multiplying the coefficients of “a” and “c”, but this time there is a more mathematical reason for doing so.  And the payoff is the same, since we then need two numbers that add to 31 and multiply to 210 — but we use the monic factoring method to perform that step.  This cements monic factoring skills, as it becomes part of the process in the later topic.  And now the miracle, as we have common factors in the two right-side terms:

6(6x^2 + 31x + 35) = (6x+10)(6x+21)

= 2(3x+5) \cdot 3(2x + 7)

= 6(3x+5)(2x+7)

And now you zap the 6 from each side and you’re done:

6x^2 + 31x + 35 = (3x+5)(2x+7)

It was shocking to me that this method works at all, and especially shocking that it always works: any factorable non-monic quadratic can be dealt with using this method.  And variable substitution is a natural method used in other places: completing the square is a variable replacement using \left(x - \frac b 2\right) as the variable … x^4 - 1 is a difference of squares … circles and ellipses all relate to the unit circle x^2 + y^2 = 1… trigonometric equations are just regular equations when you cover your hand over the “\sin x” part … a z-score is a linear substitution … and others.

The biggest benefit of presenting substitution methods as early as possible is that students learn a general-purpose tool they can apply repeatedly across grades and topics.  I also think it makes quadratic factoring easier and faster to teach.  What do you think?

Next: how this method can be used to develop the quadratic formula…

(If you know how to better display equations easily in places like this, let me know.  The LaTeX equations look pretty bad in the vertical alignment category, and I had to force a white background on each equation.  As long as it’s readable, I guess, but somehow I think it could be better.  Thanks to Mark Betnel for the pointer to the LaTeX commands available.)

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About Bowen Kerins
Bowen is one of the lead authors of CME Project, a high school mathematics curriculum focused on mathematical habits of mind. Bowen leads professional development nationally, primarily on how math content can be taught with a focus on higher-level goals. Bowen is also a champion pinball player and once won $1,000 for knowing the number of degrees in a right angle.

14 Responses to Factoring Non-Monic Quadratics

  1. Mark Betnel says:

    WordPress has a couple of latex plugins (like http://wordpress.org/extend/plugins/wp-latex/) — this one doesn’t look fantastic if you don’t have a white background in your theme, but it does work.

  2. Ashli says:

    My precalc students are sending great love your way. Just thought you should know.

    This never would have occured to me, and I also love the implications for other areas of math.

  3. R. Wright says:

    Very interesting idea. I need to think about using it, myself. But you never answered the interesting question you raised: Why do we teach this stuff?

  4. Bowen Kerins says:

    Superb question! There will be more on this, but here’s my quick take: students are learning general thinking skills that can be applied in and out of the classroom. In this case, it’s the concept of looking at something from a different perspective to gain new information. There’s different information in forms, and students learn to transform expressions with purpose and interpret the results. The “chunking” concept allows students to identify and solve simpler problems — so that when faced with a new and complex problem, they can look to simplify it to something more familiar.

    It’s my opinion, but I think “college and career readiness” is much more about these thinking skills than any specific topic. Factoring and solving equations can be areas in the curriculum where students learn a significant number of deep concepts that they can apply many times in and out of the mathematics classroom. In writing CME Project we call these the “habits of mind”, and they are similar in spirit to Common Core’s “mathematical practices”.

    Thanks for asking, and please keep asking this question! I also recommend the Phil Daro talk — he brings this up frequently.

    • R. Wright says:

      Okay, I’ll take a look at Daro’s talk tonight.

      I think the root of my problem with teaching factoring is that it’s not usually (or maybe ever) done in what I would call an open and honest way. There have been a handful of times in my own private mathematical adventures that I found it truly useful to factor an expression in order to tackle an interesting question. One field in which this tends to happen is solving differential equations — but aside from the fact that we can’t get anywhere near that level of sophistication in an algebra course, the factoring needed to solve a realistic differential equation does not fit the typical limited definition of “factoring” used in algebra, since irrational coefficients may be involved.

      Some other interesting questions that are often best addressed by factoring techniques are in number theory (even very simple number theory). Lately I’ve been wondering whether this wouldn’t be the best avenue to motivating the idea of factoring. I need to give it more thought, though.

  5. Bowen Kerins says:

    I can only speak for what we did in our curriculum, but here’s what we did:

    1. We try and get students to notice (even early in Algebra 1) that there are equations that are unsolvable by the means they are taught to solve linear equations.
    2. We ask students to solve equations like (2x + 3)(x – 5) = 0, leading to the Zero Product Property, including an analysis of why it wouldn’t work for any number but zero.
    3. We teach students that an equation like x^2 + 14x + 48 = 0 is pretty hard to solve, but if it were transformed into something like those factored ones…
    4. THEN we teach factoring because students are ready and have a purpose for the transformations they perform, and are able to solve problems that were previously inaccessible.

    It can be argued that solving quadratic equations isn’t enough motivation by itself. We give some other examples where factoring gives information about numbers: for example, the factoring of x^2 + 14x + 48 also tells me that 11,448 = 106 * 108.

    Others might have more to say, but I feel the core concepts of transformation, meaning and use of expressions, and substitution of variables are all enough to make the topic worth teaching at the Algebra 1 / 2 level… as long as those concepts are part of the discussion.

  6. alcuoco says:

    I’m not sure if this is a direct answer to the question of why we teach this stuff, but factoring sits inside of the the more general category of “transforming expressions to reveal hidden meaning.” Bowen has already given one example: factoring lets you find roots of polynomial equations.

    Sure, you can find good approximations to roots with a CAS, but if you want to make an analysis of the underlying structure of the roots, you look for factors. For example, a detailed analysis of which numbers of the form cos (2\pi/n) can be expressed in terms of iterated square roots requires that one look at the factorization of x^n-1 over the the integers and then over the complex numbers.

    Another example is the analysis of the distribution of sums when, say, 4 dice are thrown. A useful tool here is to look at various forms of

    (x+x^2+x^3+x^4+x^5+x^6)^4

    A famous problem along these lines is to find, if possible, a labeling of two dice, different from the usual one, so that the distribution of the sums is the same as what you get when you look at ordinary dice (loaded dice that act fair, in other words). This amounts to rearranging the factors (over the integers) of

    (x+x^2+x^3+x^4+x^5+x^6)^2

    under some constraints.

    The method Bowen presents for factoring non-monic quadratics works for higher degrees, too, and it allows you to transform *any* polynomial into a monic one via a change of variable. The renaissance folks used this as a key step in the development of the cubic formula.

  7. R. Wright says:

    Thanks for all the input.

  8. Pingback: Deriving the Quadratic Formula (the easy way) « Patterns in Practice

  9. chloe says:

    can anybody help me with this?
    5q^2+17q+6
    its due tomorrow, so if anyone can help you’d be a life saver!

    • Bowen Kerins says:

      Multiply through the expression by 5, noting you’ll have to pay it back later…
      25q^2 + 85q + 30

      Now “chunk” as a quadratic where “5q” is the variable:
      (5q)^2 + 17(5q) + 30

      Replace 5q by another variable, maybe H:
      H^2 + 17H + 30

      Now factor the monic: find two numbers that add to 17 and multiply to 30. These numbers are 15 and 2:
      (H + 15)(H + 2)

      Uncover the variable by replacing H with 5q:
      (5q + 15)(5q + 2)

      Pay back the 5:
      (q+3)(5q+2)

      Ta-da…

  10. Marc Garneau says:

    I really like and appreciate the conceptual integrity of this. I’ve seen ‘magic’ methods that are hailed as easy – but they’re simply procedural and not sense-making. I learned a new word too! (non-monic).
    The math geek in me couldn’t help but explore how the derivation of the quadratic formula would work out with this approach. The derivation is still quite complex, but it has the elegant benefit of completing the square with a monic quadratic instead of a non-monic one, which is conceptually more accessible.

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