The language of polynomials

Polynomials, leading term, leading coefficient, degree, monic and constant term; adding, subtracting and multiplying polynomials; identically equal polynomials; polynomial equations.

Polynomials and their parts

A polynomial is an expression of the form

$$P(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0, \qquad a_n \ne 0,$$

where every index is a whole number $0, 1, 2, \ldots$ So $3x^4 - x + 7$ is a polynomial, but $\sqrt{x}$, $\dfrac1x$ and $2^x$ are not.

Word	Meaning	For $P(x) = -2x^3 + 5x - 4$
leading term	the term with the highest power	$-2x^3$
leading coefficient	the number in front of it	$-2$
degree	the highest power	$3$ (a cubic)
constant term	the term with no $x$, which equals $P(0)$	$-4$
monic	leading coefficient $1$	not monic

Degrees $0, 1, 2, 3, 4$ are called constant, linear, quadratic, cubic and quartic. The zero polynomial $P(x) = 0$ has no leading term, so its degree is not defined.

Adding, subtracting and multiplying

Example 1●

Let $P(x) = x^3 + 2x - 1$ and $Q(x) = -x^3 + x^2 + 4$. Find $P + Q$ and the degree of $PQ$.

$P(x) + Q(x) = x^2 + 2x + 3$, degree $2$.

The x³ terms cancel.

$\deg(PQ) = 3 + 3 = 6$, with leading term $(x^3)(-x^3) = -x^6$.

Identically equal polynomials

Example 2●●

Find $a$, $b$ and $c$ if $ax^2 + (b - 1)x + 3 \equiv 2x^2 + 5x + c$.

$x^2$: $a = 2$. $\;x$: $b - 1 = 5$, so $b = 6$. $\;$Constant: $c = 3$.

Match each power of x.

Example 3●●●

Find $p$ and $q$ if $x^2 + 6x + 1 \equiv (x + p)^2 + q$.

Expand the right: $x^2 + 2px + p^2 + q$.

$2p = 6$, so $p = 3$. Then $p^2 + q = 1$, so $q = -8$.

Polynomial equations

The zeroes of a polynomial $P(x)$ are the roots of the equation $P(x) = 0$, and they are the $x$-intercepts of $y = P(x)$. Factoring is the main way to find them: $x^3 - 4x = x(x - 2)(x + 2)$ has zeroes $0, 2, -2$.

Flashcards

Degree of a product

$\deg(PQ) = \deg P + \deg Q$

Monic

Leading coefficient $1$.

Constant term

$P(0)$

$P(x) equiv Q(x)$

Equal for every $x$, so all matching coefficients are equal.

Practice

Which is a polynomial?
Find the leading coefficient of $5x - 2x^3 + 7$.
Find the degree of $(x^3 + 1)(2x^4 - x)$.
$P$ and $Q$ both have degree $3$. The degree of $P + Q$ is:
$ax^2 + (b - 1)x + 3 \equiv 2x^2 + 5x + c$. Find $b$.
$x^2 + 6x + 1 \equiv (x + p)^2 + q$. Find $q$.
Find the constant term of $P(x) = -2x^3 + 5x - 4$.
Which polynomial is monic?
How many roots does $x^3 - 4x = 0$ have?
The degree of the zero polynomial is: