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In Secondary education in Australia, the general outline for introducing techniques to solve the quadratic equation $$ x^2+bx+c=0 $$ is first to introduce the idea to find two numbers $p$ and $q$ such that $b=p+q$ and $c=pq$. After this, the quadratic formula is introduced, which is applicable to all quadratic equations.

Why are these topics this taught in this order?

It seems more natural to me to introduce the quadratic formula, which works in the general case first, and then introduce the other technique which only applicable to special cases.

Secondly, why is synthetic division not taught at all? This would allow students to extend their knowledge of solving quadratic equations to higher degree polynomials.

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    $\begingroup$ I would recommend removing both topics from the secondary curriculum, focusing instead on graphing functions and finding roots numerically. $\endgroup$ – user173 Mar 17 '14 at 14:03
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    $\begingroup$ @Chris C: Only around 10% of American college students take calculus. Preparing everyone for a class like the one you describe would not be my goal in secondary education. (The percentage is rough, maybe accurate to a factor of 2, estimated from math.duke.edu/~das/essays/renewal/students.html and nces.ed.gov/fastfacts/display.asp?id=98.) $\endgroup$ – user173 Mar 17 '14 at 15:54
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    $\begingroup$ @Matt F Calculus is one end goal and I agree only a fraction get there, but that fraction still needs to understand the entire picture. Also getting them only to graph them and algorithmically compute the roots shields them from the logical thinking that algebraically factoring gives them (though teaching both is necessary). I believe knowing how to factor isn't the end goal, but logically thinking to combine several ideas is. $\endgroup$ – Chris C Mar 17 '14 at 16:05
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    $\begingroup$ @MattF. I agree with your opinion that graphing functions and finding roots numerically should be emphasized, but I disagree with removing algebraic methods from the curriculum, and with your assessment of calculus. Even though only 10% of college students take calculus, these are the only students for whom it really matters what is in the secondary math curriculum. For most of the rest of the students, the purpose of high-school math classes is to encourage basic numeracy and promote logical thinking, and almost any selection of topics within mathematics will accomplish this goal. $\endgroup$ – Jim Belk Mar 17 '14 at 17:45
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    $\begingroup$ @MattF. The key word in the italicized statement is "really". Of course it matters to all students what the math curriculum is, but to most students it matters the same amount as the curriculum in any of their other classes. But for students who are planning to go into science, engineering, or medicine, the math curriculum can actually affect whether they will be successful in pursuing their chosen career. The math curriculum matters far more to these students than to everyone else, and this should be taken into account when deciding what to include. $\endgroup$ – Jim Belk Mar 17 '14 at 18:33
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To help students understand how we move from specific cases to the general case, I created a visual depiction of the process of completing the square. I used shapes analogous to Algebra Tiles, which are a good manipulative for building understanding of factoring as well. I then took the process one step further and illustrated the derivation of the Quadratic Formula. I think the visual is powerful in building understanding.

Completing the Square

Deriving the Quadratic Formula

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  • $\begingroup$ Great, I think that completing the square should be emphasized more often, and earlier on. I find students in calculus who have no idea how to do it, let alone appreciate its use. $\endgroup$ – Brendan W. Sullivan Mar 17 '14 at 19:02
  • $\begingroup$ And James Tanton has some good takes on this, as well: youtube.com/watch?v=OZNHYZXbLY8 $\endgroup$ – Brendan W. Sullivan Mar 17 '14 at 19:03
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To the first question, I suspect the primary reason is that the mathematical community learned to solve quadratics proceeded in this order - that is, mathematicians realized there were specific, solvable cases before proceeding to the general solution. But I think there are two related arguments to be made for maintaining that order, one historical, one practical.

The historical argument goes something like: We should teach solving quadratics in this order (special cases to the general case) because that order reflects the historical trajectory of the progress mathematicians made while working on this problem. But this argument is stronger than it looks on the surface - the reason the historical trajectory took the path that it did is because, in general, that's often what it means to do mathematics. When confronted with a new and unsolved problem, one of the ways that we make progress on the problem is to impose some assumptions on the problem (e.g. special cases) and solve the more constrained problem. Only after playing in this sandbox for a while does the general solution emerge.

The second, more practical, argument is related: by presenting the general solution up front, students are robbed of the need for it and it just becomes another formula to memorize because the teacher said so. On the other hand, by starting with the special cases, students will soon run into quadratics that do not factor. The limitations of the techniques used in the special cases can motivate the need for the general solution.

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  • $\begingroup$ Teaching <random subject> in the historical order is usually a bad idea. The logical/simple/natural development is normally quite different. Many famous theorems (e.g. Lagrange's on order of groups and subgroups) were completely different in the original setting, much more restricted. Proofs got different (e.g. Euler's theorem $a^{\phi(m)} \equiv 1 \pmod{m}$ if $\gcd(a, m) = 1$ is an almost trivial corollary to Lagrange's). $\endgroup$ – vonbrand Mar 17 '14 at 15:50
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    $\begingroup$ The second argument is certainly the more convincing one. Having to use a formula without knowing where it comes from or the reason for it is a good way to turn students off super quick. $\endgroup$ – David G Mar 17 '14 at 16:10
  • $\begingroup$ Thank you for your answer. I agree with the other comments (esp. vonbrand) that the historical order is not necessarily the best order to teach something. $\endgroup$ – Daryl Mar 24 '14 at 10:07
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Why the "find things that add and multiply" is taught before the quadratic formula

This method of "think of $p$ and $q$ so that $b=p+q$ and $c = pq$" is used in the service of factorising the polynomial equation into $(x+p)(x+q) = 0$. The concept that you can factorise and then set each part equal to zero is the fundamental idea of this method, and is in fact a fundamental idea to solving many equations, not just polynomial equations. As such, it's a part of a larger more general method that the quadratic formula is not part of. Often this section of the maths curriculum follows a section where you tell them about noticing a common factor (eg in $2x^2 + 4x = 0$ becomes $2x(x+2) = 0$), and is trying to match with this earlier idea.

The quadratic formula is usually introduced as a "last resort" when all other methods have failed. Personally, I have a sense of power when I think of a way to factorise it, rather than "resorting" to the quadratic formula. I conjecture that many people feel that using factorisation is more "righteous" than using the formula, in the same way that they feel guilty asking a computer to solve an equation for them. While you can argue on the correctness of this feeling of guilt, you can't deny it's there for a lot of people, and many of these people are teaching the students. It is probably worth noting at this point that we don't teach a "linear formula", that solves $ax + b =0$ by doing $x = -b/a$, because there are other more general methods that work just fine.

Another reason is that the concept of writing a polynomial as a product of factors is an important idea even if you are not trying to solve equations. As algebraic objects, polynomials have products and factors the way that integers do, and factoring them is a way of reducing them to their "lowest form". The quadratic formula is a way of solving an equation and is rarely used as a way to find a factorised form. Of course, there is nothing stopping you doing so -- I do find it odd that it is rare to see someone saying that they found $x = p, q$ by the quadratic formula and therefore the factorised form is $(x-p)(x-q)$!

Factors are also a way to strengthen the connection between a polynomial and its graph. Noting that you can tell the x-intercepts of a function from its formula when it is in factored form is a powerful thing. It's cool that numbers pulled out of the structure of an equation written on the page tell you about what the graph is doing. Interestingly, completing the square also strengthens this connection, because when you rewrite $y = ax^2 + bx + c$ as $y = a(x-d)^2 + e$, then you can immediately read off the location of the vertex $(d,e)$. Unfortunatley, I find it rare that students know that completing the square allows you to do this trick. (Note that we do often tell them a bit about this with linear graphs, citing the $y=mx+c$ and the $ax + by = k$ forms of a line that tell you different information.)

Why not teach (synthetic) polynomial division

I never learned synthetic division, and when I saw my South Australian friends do it when I got to Uni I found the whole thing a bit mystifying. (I may have been biased, since I already knew how to do ordinary polynomial division it seemed like a waste of time to learn another method when I had one that worked well for me, especially considering how few polynomials I really needed to divide even in my first year of a maths degree.) When I look at the first few hits of a Google search for "synthetic polynomial division", the explanations there are not particularly illuminating. It is not at all obvious how synthetic division might be used to extend understanding from quadratics to higher-degree polynomials, except for dividing by linear factors until you get a quadratic, which ordinary polynomial division does too.

So perhaps the question is really why is polynomial division (synthetic or otherwise) not taught earlier? I wish I knew. I think it would be great if the process of dividing polynomials was taught earlier. As algebraic objects it makes sense that if you can multiply polynomials then you would want to divide them too, and also it has a nice lead-in to rational functions. As you say, it would allow extension of quadratics to other functions.

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Two questions, really...

  1. If you start by considering $(x - a) (x - b)$ as a multiplication problem, later the search for factors as stated is natural when solving equations. But then you run into cases where the split isn't apparent. Enter the general formula. (That was the way I got it taught.)
  2. Beats me. In high school we learned long division of univariate polynomials. Used it little, mostly to divide out factors when solving higher degree equations.
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I teach it in high school (age equivalents 15 - 17) since it ties in so nicely to the concept of prime factoring in number theory to gain an understanding of what is making up a number. The parallel concept is what is making up the polynomial. The factoring of a polynomial lets us know about reducing algebraic fractions, performing algebraic fraction operations and for solving many quadratic equations. It is formative as it lays a foundation for understanding the general case and as shown by @Jennifer Silverman through completing the square and how there are some that produce integral and rational roots while others have irrational roots.

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The power of learning synthetic division is not that it speeds up being able to do divisions by (x-a) but as a conceptual and/or practical tool for evaluating a polynomial function f(x) at x = a. f(a) is the last "term" of the process. Often in computer science classes it is pointed out that when one wants to compute f(a) for a polynomial of high degree one does not want to compute high powers of a large number but use "synthetic" division.

I am old enough that I learned this in a Theory of Equations course but this lovely idea is not usually taught in the linear algebra courses that replaced Theory of Equations.

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  • $\begingroup$ Interesting. This is the first time anyone has given me a decent reason for synthetic division. In that case, the "division" is a bit of a misnomer isn't it? $\endgroup$ – DavidButlerUofA Sep 1 '14 at 8:00
  • $\begingroup$ Names for things are often a can of worms. You might enjoy this essay: en.wikipedia.org/wiki/Horner%27s_method $\endgroup$ – Joseph Malkevitch Sep 12 '14 at 19:30

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