Get down with the lingo
Conjugate Pair
Complex roots are always in conjugate pairs. That means that if 3 + 2i is a complex root of a function, 3 − 2i is also. Conjugate pairs are BFFs. They are always together.
Descartes' Rule Of Signs
How many positive and negative real roots are there? Check Descartes' Sign, and Burma Shave.
Discontinuity
A discontinuity is a value of x that causes a problem in a rational function. By problem, we mean that the function is not defined at that x. A discontinuity might be removable (see the definition for hole) or non-removable (see the definition for vertical asymptote.) Polynomials have no discontinuities.
End Behavior
End behavior is what a function looks like at its ends. By ends, we mean very big values of x, in the positive or negative direction.
Even-degree Polynomial
A polynomial "gets even" when its biggest exponent (on one of its variables, like x) is an even number. Its end behavior is "both arms up" or "both arms down" Sometimes called an even-ordered polynomial.
Fundamental Theorem Of Algebra
A polynomial has n roots, where n is the degree of the polynomial. Not nearly as fundamental as it wants you to think.
Hole
A removable or "cancellable" discontinuity in a rational function. A hole looks like a tiny little bald spot in a graph.
Horizontal Asymptote
Some rational functions start to look like horizontal lines once they get far away from the origin. If a rational function has a horizontal asymptote, the two of them become best buds at very big or very small values of x. The rational function is right up against that horizontal asymptote. It skates along right on top of it, or right under it.
Intermediate Value Theorem
To get from the positive to the negative parts of the graph, a function must pass zero, and must collect $200.
Lower Bound On Roots
A number that's smaller than any of the roots of a polynomial. We know that a number is a lower bound on roots if we synthetically divide with it and the signs of the third row alternate.
Odd –degree Polynomial
A polynomial whose biggest exponent on a variable is odd. The end behavior of an odd polynomial is "one arm up, one arm down."
Rational Root Theorem
A method for finding all of the possible rational roots of a polynomial. Actually finding which roots are real is another story.
Root Of Even Multiplicity
A root of even multiplicity looks like (x − 1)2 or (x – 1)100. It's a factor raised to an even power. It does not pass through the x-axis. It does not cause a change of sign. At a root of even multiplicity, a graph just taps the x-axis and bounces right back the way it came. It counts as two root, or one hundred, or whatever is its exponent.
Root Of Odd Multiplicity
A root of odd multiplicity looks like (x – 1) or (x – 1)3 (x – 1)101. It's a factor raised to an odd power. It passes right through the x-axis. A function changes sign at a root of odd multiplicity. It counts as three roots, or one hundred and one, or whatever is its exponent.
Slant Asymptote
When the degree of a rational function's numerator is exactly one larger than it's denominator, we get slanted asymptotes that the graph may or may not cross.
Upper Bound On Roots
A positive number that's bigger than any possible root of a polynomial. If we synthetically divide with a number and the third row has all the same sign, we know that number is an upper bound on roots.
Vertical Asymptote
For a rational function, a value of x that would make its denominator zero. The rational function would be undefined at that x. On a graph, it looks like a vertical electric fence. A rational function will "whoosh" up or down the side of a vertical asymptote. It will never touch the vertical asymptote. It is a non-removable discontinuity.