Chapter 6
Polynomial Division and Roots
Seventeen divided by five is three, with two left over. You have been splitting numbers that way since long before algebra. A quadratic can be factored by hand, and the quadratic formula finishes anything it cannot, but a cubic offers nothing to guess at and no formula that friendly is waiting. Polynomials can be divided by one another exactly the way whole numbers can, remainder and all. The leftover piece turns out to say something about the polynomial that nobody would expect from a division problem.
What You'll Explore
5 lessons.
- Polynomials and Their Graphs
Sketch a few polynomial graphs and their shapes are clearly not arbitrary. How far can one wander, how many times can it cross the horizontal axis, and where does it head far out to the left and right? You will start here, with what these expressions are, before anything gets divided.
- Polynomial Long Division
Long division of whole numbers has a rule for when to stop: keep going until what is left is smaller than the divisor. Polynomials have no size in that sense, so what does smaller even mean here? This lesson rebuilds the procedure from scratch with that question at its center.
- Synthetic Division and the Remainder Theorem
Most of the work in a long division problem is copying terms you have already written. When the divisor is as simple as x minus a number, nearly all of that writing can go. The whole calculation collapses into one short row, and the final number in it answers a question that has nothing to do with dividing.
- The Factor Theorem
Suppose you spot a single number that makes a cubic equal zero. Does that one fact help you break the whole expression apart, and could the process run the other way, building a polynomial from the roots you want it to have? Those two directions are what this lesson is about.
- The Rational Root Theorem
Everything so far has assumed you were handed a root to work with. Where would the first one come from? Testing every fraction is hopeless, but a polynomial with whole number coefficients quietly restricts which fractions could possibly work. Finding that restriction, and seeing what it still leaves you to check, is the work here.