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Precalculus: An Investigation of Functions, by David Lippman and Melonie Rasmussen, licensed under CC BY-SA 4.0. Read the original free This shelf’s study material is shared under the same license (CC BY-SA 4.0).
Chapter 2 · pp. 6–6
This chapter pulls together several core precalculus ideas: how to find and use zeros of polynomials, how rational functions behave for large inputs, how vectors combine through the dot product, how conic sections arise, and how trigonometry links triangles to the unit circle. Each idea comes with a short example so you can try the reasoning yourself.
Polynomial long division works like long division with numbers. You repeatedly remove the highest-degree term of what is left. The steps are:
Example: . Dividing by gives . Multiply: . Subtract to get . Dividing by gives . Multiply: . Subtract to get 0. The quotient is with remainder 0.
The factor theorem says that is a factor of a polynomial exactly when . The reason is that if , dividing by leaves remainder 0, so the division is exact. Example: for , we get . So is a factor, and indeed .
The two ideas work together. Once you find a zero , you can divide by to get a simpler polynomial and keep going.
The rational roots theorem gives a finite list of candidates. If a polynomial has integer coefficients, any rational zero has the form , where divides the constant term and divides the leading coefficient.
Example: for , is among and is among . The candidates are . Testing shows that is a zero, since .
Common mistake: thinking the theorem finds every zero. It only lists possible rational zeros, and each one must still be tested. Irrational and nonreal zeros are never on the list. For , the candidates are , and none works. Its zeros are .
Conjugate pairs. If a polynomial has real coefficients and is a zero with , then is also a zero. Nonreal zeros come in pairs. For example, has zeros and . A consequence: since nonreal zeros pair up, an odd-degree polynomial with real coefficients must have at least one real zero.
A rational function has an oblique (slant) asymptote when the degree of the numerator is exactly one more than the degree of the denominator. Dividing the numerator by the denominator gives a linear quotient plus a remainder term. That remainder term shrinks toward 0 when is large, so the graph gets close to the line given by the quotient.
Example: . As grows, approaches 0, so the graph approaches the line . Polynomial long division is the tool you use to find the line when the division is not as easy to see.
The dot product of vectors and is
where is the angle between the vectors. The component form is easy to compute. The cosine form connects the result to the angle.
Example: .
If the dot product is 0, then (for nonzero vectors), so the vectors are perpendicular. You can also solve the cosine form for to find the angle between two vectors.
Common mistake: believing that the dot product is a vector. It is a single number (a scalar), found by multiplying matching components and adding the products. For example, , which is a number, not a vector. (It is 0 because these vectors are perpendicular.)
Conic sections are the curves formed when a plane slices a double cone. The four types are the circle, ellipse, parabola, and hyperbola. The angle of the slice decides which curve you get:
Right-triangle trigonometry and unit circle trigonometry describe the same functions. On the unit circle, the point at angle has coordinates . So is the x-coordinate and is the y-coordinate.
For an acute angle, picture a right triangle with hypotenuse 1 inside the unit circle. Its legs have lengths and . This matches the ratios adjacent/hypotenuse and opposite/hypotenuse, because dividing by a hypotenuse of 1 changes nothing.
The unit circle goes further. It lets you define the functions for angles beyond , where no right triangle contains the angle. Example: , the y-coordinate of the point at .
