Numbers & Functions

4 Polynomials, Factoring, and the Quadratic Formula

CHAPTER 4 Polynomials, Factoring, and the Quadratic Formula You now know what a function is — a reliable machine, with a domain and a range, that you can graph, chain, and sometimes reverse. From here on we meet the famous machines, the ones that show up so often they’ve earned names. This chapter is about the simplest and friendliest family of all.

They’re so simple you can build every one of them using nothing but the operations you’ve trusted since Chapter 1: adding, subtracting, and multiplying. No division, no roots, no exotic ingredients. These are the polynomials.

4.1 What a polynomial is

A polynomial is any machine you can write as a sum of terms, where each term is a number times x raised to a whole-number power. That sounds stuffy, so look at one: f(x) = 3x2 + 2x −5 Three terms. The first is 3 times x2. The second is 2 times x (which is x1). The last is −5, which you can think of as −5 × x0, since anything to the power 0 is 1. Every term is a number times a whole-number power of x. That’s the whole definition.

And here are some things that are not polynomials, so you can feel the boundary: √x (that’s x to the power 1 2 — not a whole number), 1 x (power −1 — negative), and 2x (the x is up in the exponent — a completely different beast, the next chapter’s).

Because they’re built only from multiplying and adding, polynomials never jam: Every polynomial accepts every real number.

Its domain is always (−∞, ∞).

4.2 The vocabulary (just two words)

The degree of a polynomial is its highest power. 3x2 + 2x −5 has degree 2; x3 −2x2 + x −8 has degree 3; plain 7 has degree 0. Degree controls the machine’s whole personality. The leading term is the term with that highest power. When x gets huge, the leading term bullies all the others into irrelevance, so it alone decides what the machine does far out at the edges. Degrees have nicknames: degree 1 is linear, degree 2 is quadratic, degree 3 is cubic.

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