A packaging designer cuts equal squares from the corners of a sheet of cardboard and folds up the sides. A business owner notices that raising the price brings in more money, up to a point, and then less. In both cases the quantity that matters — a volume, a revenue — is a polynomial in one variable, and the questions are the ones this unit has been preparing you for: where is it zero, where is it positive, when does it hit a target value, and where is it largest? Before building those models, it helps to see how much a polynomial's zeros tell you, and how little extra information is needed to pin down the rest.
A family of polynomials
Write down three zeros, say , and . The cubic has those zeros, but so does , and so does . Multiplying by a nonzero constant stretches the graph vertically (and reflects it if ) but cannot move a point where , because . So the zeros define a whole family of polynomials, one member for each value of . To pick out one member you need one more piece of information — any point on the curve that is not a zero. Substitute its coordinates and solve the resulting linear equation for . Repeated zeros carry their multiplicity into the family, and irrational zeros come in conjugate pairs such as , whose two factors multiply to a quadratic with integer coefficients.
Even and odd symmetry
Some polynomials are symmetric, and spotting it halves the work of graphing. A function is even if for every : its graph is a mirror image across the -axis. A function is odd if : rotating its graph about the origin leaves it unchanged. For polynomials there is a quick test. Replacing with leaves every even power unchanged and flips the sign of every odd power. So a polynomial with only even powers (a constant counts, since is even) is even, a polynomial with only odd powers is odd, and a mixture is neither. Always confirm with the algebraic test rather than by eye, because a graph can look nearly symmetric without being so.
| Function | Compute f(−x) | Classification | Graph |
|---|---|---|---|
| Even | Symmetric in the y-axis; zeros ±1, ±2 | ||
| Odd | Symmetric about the origin; , | ||
| Neither | Equals neither h(x) nor −h(x) | ||
| Neither | The constant term is an even power, so it spoils oddness |
Polynomial models from real situations
Take a cm by cm sheet of cardboard and cut an cm square from each corner. Folding up the flaps gives a box of height , length and width , so its volume is a cubic. The model only makes sense where every dimension is positive, so the domain is , even though the polynomial itself is defined everywhere. Asking for a particular volume turns into a polynomial equation: subtract the target, divide out any common factor, then use the factor theorem to find a rational root and factor the rest. Asking for the largest volume is a different kind of question, because no factoring finds a maximum; for now you use graphing technology to locate the highest point in the domain. Revenue works the same way. If a company's revenue is thousand dollars when it sells thousand units, factoring shows revenue is positive only for , and setting equal to a target gives a cubic you can solve by the factor theorem.
| Situation | Model | Domain from the context | Typical question |
|---|---|---|---|
| Open box from a sheet, cut | Which cut gives a set volume? Which gives the most? | ||
| Revenue at thousand units | When is revenue positive? When does it reach a target? | ||
| Prism whose length is , width , height | What value of x gives a required volume? | ||
| Ball height seconds after launch | until | When does it land? When is it above a given height? |
Graphing technology answers the questions algebra cannot. Enter the model, set a window that matches the domain, and use the zero, intersect and maximum features. To solve graphically, graph and and find where they intersect; to find the largest volume, use the maximum feature on the part of the curve inside the domain. Technology gives decimal approximations, so report them to a sensible precision — a cut to the nearest hundredth of a centimetre is already more precise than scissors — and, where exact roots exist, find them algebraically and use the graph as a check. A graph also catches algebra slips: if your factored answer claims a root at but the curve clearly does not reach there, one of them is wrong.