Polynomials, degree, and notation

Before we hunt for zeroes we need to be precise about what a polynomial is. The English word polynomial literally means "many terms", but only certain kinds of terms are allowed.

Definitions

A polynomial in $x$ is an expression of the form $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0,$ where $n$ is a non-negative integer, the coefficients $a_n, \ldots, a_0$ are real numbers, and $a_n \ne 0$ . The number $n$ is the degree of the polynomial, $a_n$ is the leading coefficient, and $a_0$ is the constant term.

The most common families are:

Linear polynomial , degree $1$ : $p(x) = ax + b$ with $a \ne 0$ .
Quadratic polynomial , degree $2$ : $p(x) = ax^2 + bx + c$ with $a \ne 0$ .
Cubic polynomial , degree $3$ : $p(x) = ax^3 + bx^2 + cx + d$ with $a \ne 0$ .
Constant polynomial , degree $0$ : $p(x) = c$ with $c \ne 0$ . (The polynomial $p(x) = 0$ is the zero polynomial and its degree is undefined.)

A zero (or root) of $p(x)$ is a value $k$ such that $p(k) = 0$ .

Theorem / Concept: degree and zeroes

A first basic fact: a polynomial of degree $n$ has at most $n$ (real) zeroes. A linear polynomial has exactly one zero. A quadratic has at most two real zeroes , and can have zero, one (a "repeated" or "double" root), or two distinct real zeroes. A cubic has at most three real zeroes; it must have at least one real zero (because its graph runs from $-\infty$ to $+\infty$ , so it must cross the $x$ -axis at least once).

We will see this geometrically in the next topic: the zeroes are precisely the $x$ -coordinates where $y = p(x)$ meets the $x$ -axis.

Be careful about what is and isn't a polynomial. Allowed: $3x^2 - 5x + 1$ , $x^4 - x$ , $-7$ . Not allowed: $\sqrt{x}$ (fractional power), $x + 1/x$ (negative power), $|x|$ (not algebraic). Only non-negative integer powers of $x$ , combined linearly with real coefficients.

Also be clean about degree. The degree is the largest exponent that appears with a non-zero coefficient. For $p(x) = -7 + 4x^3 + 0 \cdot x^5$ , the degree is $3$ , not $5$ . Always tidy up the polynomial , combine like terms , before reading off the degree.

We will write $\alpha$ , $\beta$ (and sometimes $\gamma$ ) for zeroes throughout this chapter; these letters became standard in classical algebra and you will meet them again in trigonometry.

Worked examples

Example 1. Identify each as linear, quadratic, cubic, or none: (a) $3x - 7$ , (b) $x^2 + 2$ , (c) $5x^3 - x + 1$ , (d) $\sqrt{x} + 1$ , (e) $0$ , (f) $\dfrac{1}{x} + 2$ .

(a) Linear (degree $1$ ). (b) Quadratic. (c) Cubic. (d) Not a polynomial (fractional power). (e) The zero polynomial; not classified as linear/quadratic/cubic. (f) Not a polynomial (negative power).

Example 2. Find the zero of the linear polynomial $p(x) = 2x - 5$ .

Set $p(x) = 0$ : $2x - 5 = 0 \implies x = 5/2$ .

Example 3. Verify that $x = 2$ is a zero of $p(x) = x^3 - 4x^2 + 3x + 2$ .

$p(2) = 8 - 16 + 6 + 2 = 0$ . So yes, $x = 2$ is a zero.

Example 4. Find the value of $k$ if $x = -1$ is a zero of $p(x) = x^2 + kx + 6$ .

$p(-1) = 1 - k + 6 = 0$ , so $k = 7$ .

Example 5. Is $1$ a zero of $p(x) = x^4 - 3x + 2$ ?

$p(1) = 1 - 3 + 2 = 0$ . Yes.

Try it yourself

Classify by degree: $4 - x^2$ , $x^3 - x + 7$ , $2x + 5$ , $5x^4$ , $x + \dfrac{1}{x}$ .
Find the zero of $p(x) = 3x + 4$ .
Verify whether $x = -2$ is a zero of $p(x) = x^3 + 3x^2 - 4$ .
Find $k$ so that $x = 2$ is a zero of $p(x) = x^2 - kx - 4$ .
Write a quadratic polynomial whose constant term is $-6$ and leading coefficient is $1$ , with a zero $x = -3$ .
State, with reason, the degree of $p(x) = 0 \cdot x^3 + 2x^2 - 7$ .
List all rational zeroes of $p(x) = 2x - 7$ .
Give an example of a cubic polynomial with exactly one real zero.
If $p(x)$ has degree $4$ , what is the maximum number of real zeroes possible? Minimum?
Find $k$ if both $x = 1$ and $x = 2$ are zeroes of $x^3 + k x^2 + 11 x - 6$ .

Pitfalls / Insight

Coefficient zero hides the degree. $4x^2 + 0 \cdot x^4 - 1$ is quadratic, not quartic.
A polynomial must allow only non-negative integer powers of $x$ . $\sqrt{x}$ and $1/x$ are out.
A degree- $n$ polynomial has at most $n$ zeroes , never more.

Insight. A polynomial is a recipe , substitute any number for $x$ and you get an output. The interesting inputs are those for which the output is $0$ . The whole rest of this chapter is about hunting those inputs cleverly.