In this assignment, we will study rational polynomial equations. A polynomial involves numbers and powers of a variable x. The general definition we'll use is a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 , where a_n does not equal zero, and all a_i are real numbers. By restricting the first coefficient to non-zero values, we avoid the problem of dealing with varying numbers of zeroes at the front.

Sometimes we refer to a polynomial as a polynomial function, f(x) or g(x). The degree of a polynomial is n, the highest power with a nonzero coefficient. A nonzero constant has degree 0. (The zero polynomial, f(x) = 0, is a special case with no degree.) A linear polynomial has degree 1, in the form a_1 x + a_0 . A root of a polynomial is a value r such that when we substitute for x, we get zero. In function notation, root f(r) = 0.

A rational polynomial equation sets y to be the result of dividing one polynomial function by another polynomial function. Given f(x) and g(x), we can graph this in the coordinate plane (x, y).

y = \frac{f(x)}{g(x)}

The case where f(x) and g(x) both have degree zero just sets y = \frac{f}{g} , making y equal a constant. That's a horizontal line. If we increase the degree of f(x), while keeping the degree of g(x) at zero, we get y = \frac{a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0}{g} , which just divides a function by a constant. The study of basic functions is very important, but not the goal here. We will examine cases where the denominator has positive degree.

We'll focus on the smallest degree case, where both the numerator and denominator have degree 1. Instead of carrying around subscripts, we'll define the coefficients as a, b, c, and d.

y = \frac{ a x + b }{ c x + d }

For example, here's the graph with a = 3, b = 3.5, c = -1.8, d = 2. The blue line indicates the value of the function. It stays in the upper left and lower right corners of the graph. I also drew two dashed lines on the graph, which we'll discuss later.

Before we move into algebra, I suggest that you experiment with various values of a, b, c, and d. I created a GeoGebra applet where you can move sliders and see what happens. In particular, try to make the blue line flip, so it travels around the lower left and upper right corners.

The applet is on the stretch.html page.

One of the basic rules of arithmetic is that division by zero is never allowed. We don't panic - we just can't do that. In our equation, the denominator equals zero whenever x is a root. Because the denominator is linear, the Fundamental Theorem of Algebra notes that there can be only one possible root. (The theorem actually is stronger; see the notes at the bottom for more.) We can solve for that root.

{ c x + d } = 0 \rightarrow x = \frac{-d}{c}

In the applet, the purple vertical line, labeled DivZero, represents the root. DivZero is known as a vertical asymptote, because the function can come very, very close, yet never touch. As x approaches the root, the denominator becomes very, very small. The fraction therefore becomes very, very big, in the positive or negative direction. We need to consider that direction.

As noted, sometimes the graph lies in the upper left and lower right, while other times the curve is in the lower left and upper right. Let's start with an example. We'll set b = 1, c = -1, d = 2. The DivZero line is at x = \frac{-2}{-1} = 2 . When a = -2, the graph lies towards the lower left and upper right.

On the other hand, when a = 2, the graph switches corners.

We need to examine the numerator and denominator near the asymptote. We know the denominator will be near zero, but the sign is important. We want to look at the value when x is just a little more than the asymptote, say 0.01 more. For instance, with c = -1 and d = 2, g(x) = {-1} x + 2 . We found the asymptote at 2, so we find g(2.01) = {-1} (2.01) + 2 = -0.01 . The sign is negative.

Then, we evaluate the numerator at the same point. With a = 2 and b = 1, f(2.01) = {2} (2.01) + 1 = 5.02. With a positive numerator and negative denominator, y is negative.This means when we move out from the DivZero line towards positive numbers, the blue y line rises up from negative infinity. We're in the lower right corner.

If we did this from the other side, setting x a little less than DivZero, we'd find that when we move out from DivZero in the negative direction, the blue line drops down from positive infinity, in the upper left.

As we've seen, other cases flip the corners. For our example with f(x) = -2 x + 1, f(2.01) = {-2} (2.01) + 1 = -3.02. y has a negative numerator and negative denominator, making a positive ratio. We drop down from positive infinity into the upper right - and the line will also be in the lower left.

In general, we can determine the corners through this process (though there are also other methods). Find the asymptote line DivZero x = \frac{-d}{c} . Then, evaluate the numerator and denominator at a value slightly more than DivZero.

If the signs are the same, the line will drop from positive infinity and will be in upper right and lower left.

If the signs are different, the line will rise from negative infinity and will be in lower right and upper left.

The blue line also never crosses the pink horizontal line, but it looks like it comes close, just like the purple vertical asymptote. As you might suspect, the pink line is also an asymptote. To find its value, we don't need to think about zero; we need to head in the other direction and think about infinity.

For very large and very small values of x, the values of b and d only matter a little. In f(x) = ax + b, the multiplier on a becomes much more important than b. Similarly, for g(x) = cx + d, coefficient c is priviledged over d. This yields an approximate value for y.

y = \frac{ a x + b }{ c x + d } \approx \frac{ax}{cx} \approx \frac{a}{c}

In GeoGebra, I drew that horizontal line in pink, labeling it InfLim. It makes sense, because it is the limit at infinity.

When we move to higher degrees, we still can consider zeroes and behavior near infinity, but it gets more complicated. Vertical asymptotes are still determined exclusively by the denominator. We have to look for roots, which will be vertical asymptotes. For example, I graphed the rational fraction y = \frac{0.9 x^2 + 0.2 x}{x^2 + 5 x + 2} . The denominator has degree 2, so it has up to 2 real roots. The quadratic formula provides those roots, x = \frac{-5 \pm \sqrt{5^2 - 4 (1)(2)} }{2 (1)} = \frac{-5 \pm \sqrt{17} }{2} . I added in the two asymptote lines. It might look like the function touches the asymptote near zero, but it doesn't; it's just really close.

Dealing with infinity is more complex when the degrees expand. Like the linear case, only the highest powers matter. Let's say the numerator f(x) has degree n and the denominator g(x) has degree d, so we're comparing f_n x^n and g_d x^d . There are several cases.

If d > n, when we divide \frac{f_n x^n}{g_d x^d} , an x power remains in the denominator but not the numerator. Since we have a constant times \frac{1}{x^{(d-n)}} , as we head to infinity we head towards zero. Signs don't matter, since zero is zero. The sample graph below plots y = \frac{2.5 x - 4}{x^2 + 5 x + 2} .

If d < n, when we divide \frac{f_n x^n}{g_d x^d} , we keep an x power in the numerator but not the denominator. We need to consider signs in \frac{f_n x^{(n-d)}}{g_d} , because y can reach negative or positive infinity. For example, if we consider y = \frac{2 x^2 - 2.5 x}{3 x + 2} , the signs of f_n = 2 and g_d = 3 are both positive. This means for large positive x, we head towards really big positive y, while for negative x, we slide downwards.

Flip a sign, to -2 x^2 , and the edge directions reverse.

Finally, when d = n, the powers of x cancel. Like the linear case, the value of y approaches a constant nonzero limit in both directions, \frac{f_n}{g_d} . Here, I graphed y = \frac{2 x^2 - 2.5 x}{1.5 x^2 + 3 x + 2} , making the infinity limit \frac{2}{1.5} = \frac{4}{3} . (Also, because the denominator has complex valued roots, we never divide by zero and there are no vertical asymptotes.)

There's much more we could explore with rational equations, but this is a good beginning. Let's head off to a new house to play - in this case, a square inside a semicircle.

In the main text, the Fundamental Theorem of Algebra was used to state that a polynomial of degree n has no more than n real number roots. More precisely, the theorem states that any degree n polynomial, n > 0, can be factored into a multiplication of exactly n polynomials of degree 1, ( x - r_i ) , where the r_i are complex numbers and can repeat. Complex numbers are more complicated than real numbers, so we avoided them in the main text. Many proofs of the theorem are available, though the first accurate one was only about 200 years ago. Cut-the-knot has a page of them.

The section headings are mostly references to Toy Story and its sequels.