Polynomials and the Binomial Theorem

Vocabulary flashcards covering polynomials, factoring, root theorems, equation solving, and the Binomial Theorem.

Polynomial in $x$

A function of the form $a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0$ where $n \in \mathbb{Z}_{\ge 0}$, $a_n, a_{n-1}, \dots, a_2, a_1, a_0 \in \mathbb{R}$ or $\mathbb{C}$, and $a_n \neq 0$.

Degree of a Polynomial

The highest exponent $n$ appearing in a polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_0$ with $a_n \neq 0$.

Leading Term

The term $a_n x^n$ in the polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$.

Leading Coefficient

The coefficient $a_n$ associated with the highest power term in a polynomial.

Root (or Zero) of a Polynomial

A (complex) number $\alpha$ such that $f(\alpha) = 0$.

Factor Theorem

Let $f(x)$ be a polynomial and $c \in \mathbb{C}$. Then $c$ is a root of $f(x)$ if and only if $x - c$ is a factor of $f(x)$ (meaning $f(x) = (x - c)q(x)$ for some polynomial $q(x)$).

Rational Root Theorem

Let $f(x) = a_n x^n + \dots + a_1 x + a_0$ be a polynomial with integer coefficients. If $\frac{p}{q}$ is a rational root of $f(x)$ where $p$ and $q$ are relatively prime integers, then $p$ divides $a_0$ and $q$ divides $a_n$.

Quadratic Factorization Formula

If the quadratic polynomial $a x^2 + b x + c$ has roots $x_1$ and $x_2$, then it factors completely as $a x^2 + b x + c = a(x - x_1)(x - x_2)$.

Conjugate Root Theorem

Let $f(x)$ be a polynomial with real coefficients. Suppose $a + b i$ is a root of $f(x)$ where $a, b \in \mathbb{R}$ and $b \neq 0$. Then $a - b i$ is also a root of $f(x)$.

Multiplicity of a Root

The largest power of $x - \alpha$ in the factorization of a polynomial $f(x)$, where $\alpha$ is a root of $f(x)$.

Fundamental Theorem of Algebra

A theorem stating that a polynomial $f(x)$ of degree $n$ has exactly $n$ (possibly complex) roots when counting multiplicities.

Relations Between Roots and Coefficients (Quadratic)

For a quadratic polynomial $f(x) = a x^2 + b x + c$ with roots $x_1$ and $x_2$, the sum of the roots is $x_1 + x_2 = -\frac{b}{a}$ and the product of the roots is $x_1 x_2 = \frac{c}{a}$.

Factorial ($n!$)

For $n \in \mathbb{N}$, defined as $n! = 1 \cdot 2 \cdot 3 \cdots (n - 1) \cdot n$. By definition, $0! = 1$.

Binomial Coefficient

For $0 \le k \le n$, the quantity defined as $\binom{n}{k} = \frac{n!}{k!(n - k)!}$ (also written $C_n^k$, pronounced 'n choose k'), which is always a natural number.

Pascal's Triangle

A triangular array where for each row $n = 0, 1, 2, \dots$, the entries are the binomial coefficients $\binom{n}{k}$ for $k = 0, 1, \dots, n$.

Binomial Theorem

For all $n \in \mathbb{Z}_{\ge 0}$ and all $a, b$, $(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \dots + \binom{n}{k} a^{n-k} b^k + \dots + \binom{n}{n} a^0 b^n$.

General Term of a Binomial Expansion

The term given by $\binom{n}{k} a^{n-k} b^k$ in the expansion of $(a + b)^n$, which contains a total of $n + 1$ terms.