Algebra 2

Rational Expressions & Polynomial Division

A rational expression is one polynomial divided by another — a fraction where the top and bottom are polynomials. These create some of the most dramatic and interesting graphs in algebra: curves that shoot off to infinity, have gaps, and approach lines they never quite reach.

Part 1: The Simplest Rational Function

The most basic rational function is:

f(x) = \frac00

Notice the two key features:

At x = 0, the function is undefined (you can’t divide by zero!) — the graph has a vertical asymptote
As x gets very large (positive or negative), the function approaches 0 — that’s a horizontal asymptote

Part 2: Vertical Asymptotes — Where Things Blow Up

A vertical asymptote occurs wherever the denominator equals zero (and the numerator doesn’t also equal zero at that point).

Let’s explore f(x) = 1/(x - d). The asymptote moves with d:

Asymptote position (d)2

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f(x) = \frac0{x - 2}

Try This

Drag d around. The vertical asymptote follows! The function explodes to positive or negative infinity as x approaches d from either side. The denominator becomes tiny (nearly zero), making the fraction huge.

Part 3: Adjustable Rational Function

Now let’s look at a more general form. A linear-over-linear rational function:

f(x) = \frac{ax + b}{x + c}

Numerator a1

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Numerator b2

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Denominator c1

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f(x) = \frac{1x + 2}{x + 1}

Connection

Key observations:

The vertical asymptote is at x = -c (where the denominator is zero)
The horizontal asymptote is y = a (the ratio of leading coefficients)
The y-intercept is f(0) = b/c
Changing b shifts the curve without moving the asymptotes

Part 4: Quadratic Numerator — Two Roots

When the numerator is a quadratic, the function can cross the x-axis up to twice:

Root 1 (p)1

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Root 2 (q)-2

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Asymptote position0

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f(x) = \frac{(x - 1)(x - -2)}{x - 0}

Try This

Play with the positions:

The function crosses zero at x = p and x = q (the numerator’s roots)
The vertical asymptote is at the denominator’s root
What happens when you move a root to match the asymptote? Try setting p equal to the asymptote position!

Part 5: Holes vs. Asymptotes

Here’s a critical distinction. When a factor cancels between numerator and denominator, you get a hole (removable discontinuity), not an asymptote.

Consider: f(x) = (x - h)(x + 1) / (x - h). The (x - h) cancels!

Hole position (h)2

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f(x) = \frac{(x - 2)(x + 1)}{x - 2} = x + 1 \text{ (with a hole at x = h)}

Connection

Hole vs. Asymptote:

Hole: A factor cancels. The graph looks like the simplified version with a single missing point. No explosion to infinity.
Asymptote: A factor does NOT cancel. The graph blows up to infinity.

In the graph above, the blue curve looks exactly like y = x + 1, but with an invisible hole at x = h. Move h to see the hole travel along the line.

Part 6: Polynomial Long Division — Finding Slant Asymptotes

When the numerator has a higher degree than the denominator, polynomial division reveals a slant (oblique) asymptote:

\frac{x^2 + bx + c}{x - d} = (x + \text0) + \frac{\text0}{x - d}

b (numerator)0

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c (numerator)-2

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d (denominator root)1

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f(x) = \frac{x^2 + 0x + -2}{x - 1}

Challenge

Challenge: For f(x) = (x^2 - 4) / (x - 1):

Where is the vertical asymptote? (Set d = 1, b = 0, c = -4)
What is the slant asymptote? (Perform the division!)
Where does f(x) cross the x-axis? (Set numerator = 0: x^2 - 4 = 0)

The slant asymptote is approximately y = x + d + b. Verify it on the graph!

Wrapping Up

Feature	How to Find It
Vertical asymptote	Set denominator = 0 (if factor doesn’t cancel)
Horizontal asymptote	Compare degrees: same degree = ratio of leading coefficients
Hole	Factor that cancels between numerator and denominator
x-intercepts	Set numerator = 0
Slant asymptote	Polynomial long division when degree(num) = degree(den) + 1

Rational functions combine the behavior of polynomials with the drama of division by zero. Master their asymptotes and holes, and you can sketch any rational function by hand.

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