Rational Roots

Continuing in the project to blog the major topics in the Pre-Calculus curriculum (as I see them) brings us to finding the rational roots of polynomial functions.

The topics involved in studying the Rational Roots Theorem seem to me to fall into two major categories – polynomial factorization and the analysis of polynomial functions and their roots.

More after the break...

Although numerical long division has fallen out of favor in some educators' minds, it's very helpful for students to have an understanding of the standard process of long division in order to better understand polynomial division. As Hung-Hsi Wu pointed out at some point, what's really important in long division is that being given two numbers a and b, the numbers q and r can satisfactorily be determined so that a=bq+r.

In other words, find the quotient and the remainder. I still remember checking my answers to long division problems in elementary school by multiplying the quotient by the divisor and adding the remainder. This is at the heart of the division algorithm.

The same is true in polynomial division – what's important is finding the quotient and the remainder. I typically teach polynomial long division as a process similar to numerical long division. I tend not to focus too much on it and proceed pretty quickly to synthetic division but I don't feel that I'm skimping on the topic because many textbooks either don't have problems to practice long division at all or only give simple problems that can be solved by synthetic division. I think that this approach is too streamlined and obscures the foundational aspects of the concept.

Finding the roots of polynomial functions involves finding the x-intercepts of the curve. If these x- intercepts are rational numbers, then the polynomial will have linear factors with integer coefficients. A polynomial might have other factors that aren't linear, and these would produce irrational or complex roots.

Here, the topic touches on the corollary to the Fundamental Theorem of Algebra – that there may be as many roots as the degree of the polynomial, but never more. If there are fewer real roots than the degree of the polynomial, then there may be repeated roots or complex roots.

There are a lot of ideas to wade around in here (including Newton's Method), so to focus, I typically cover:

Polynomial Long Division
Synthetic Division
Determining Linear Factors given rational roots
Determining Roots given a polynomial

Here is a link to some problems (w/answers) I typically assign for these topics.

We generally use the TI-83/84 in our 100-sequence courses (College Algebra, Pre-Calculus and Trigonometry), so I have the students find the roots on the calculator and then use synthetic division to factor a cubic or quartic polynomial into linear and quadratic factors. The quadratic factors often have complex roots. In different circumstances, I would also cover the Rational Roots Theorem which is a pretty nifty method for finding the roots algebraically.

The Sullivan text covers Synthetic Division in Section R.7, with scant discussion of polynomial long division and then covers the Rational Roots material in Sections 5.5 – 5.6. Coburn offers a similar treatment of long and synthetic division and rational roots in Sections 3.2 – 3.3.

The Brown/Robbins book doesn't cover polynomial long division (I believe this is covered in Algebra Structure and Method book 2, so the author assumes it as a prerequisite), but does cover synthetic division and the Rational Roots theorem in Sections 2-5 and 2-6.

The Dolciani text covers these topics in sections 6-5 through 6-9 and includes some of the “old school” material on bounds for the real roots and Descartes rule of signs.

The Sobel text covers polynomial long division in section 1.9, then picks up with synthetic division and the rational roots theorem in Sections 3.6 and 3.7

Foerster gives a somewhat different approach than the others in his Precalculus text in Sections 16-1 through 16-5.

Each textbook covers these topics a little bit differently. One of the foundations of my teaching approach has always been to compare different textbooks to see how each one covers a particular topic and then draw from all of them to build a unit of material that is appropriate for the student population I'm working with.

I found two questions from my SAT Math Level 2 Subject Test prep book that relate to these topics. They are at a considerably lower level of difficulty than the question from the Japanese University Entrance Exam I posted a few weeks ago.

If 3 and -2 are both zeros of the polynomial p(x), then a factor of p(x) is

a) x^2 – 6
b) x^2 – x – 6
c) x^2 + 6
d) x^2 + x – 6
e) x^2 + x + 6


If f is a polynomial of degree 3, four of whose values are shown in the table below, then f(x) could equal

a) (x+1/2)(x+1)(x+2)
b) (x+1)(x-2)(x-1/2)
c) (x+1)(x-2)(x-1)
d) (x+2)(x-1/2)(x-1)
e) (x+2)(x+1)(x-2)

x f(x)
-1 0
0 1
1 -1
2 0