Polynomials - Polynomials in One Variable

A polynomial in one variable $x$ is an algebraic expression of the form $p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $n$ is a whole number and $a_n, a_{n-1}, \dots, a_0$ are constants. Visually, imagine a chain of terms where each link consists of a numerical coefficient and the variable $x$ raised to a non-negative integer power.

Each part of a polynomial separated by '+' or '-' signs is called a 'term'. For every term, the numerical factor is called the 'coefficient'. For example, in $3x^2 - 5x + 7$, the terms are $3x^2$, $-5x$, and $7$, while the coefficient of $x^2$ is $3$. Visually, coefficients represent the 'magnitude' or 'scale' of each power of $x$ in the expression.

Polynomials are categorized by the number of terms they contain: a 'Monomial' has exactly one term (e.g., $5x^3$), a 'Binomial' has two terms (e.g., $2x + 1$), and a 'Trinomial' has three terms (e.g., $x^2 + 3x + 2$). Visually, these can be thought of as a single block, a pair of blocks, or a triplet of algebraic components joined together.

The 'Degree' of a polynomial is the highest power of the variable in that polynomial. For a non-zero constant polynomial like $7$, the degree is $0$ because it can be written as $7x^0$. The degree of the 'zero polynomial' ($0$) is not defined. Visually, the degree indicates the complexity of the polynomial's behavior; the higher the degree, the more potential 'curves' or 'turns' the graph can have.

Based on the degree, polynomials are classified as: 'Linear' (degree $1$, e.g., $ax + b$), 'Quadratic' (degree $2$, e.g., $ax^2 + bx + c$), and 'Cubic' (degree $3$, e.g., $ax^3 + bx^2 + cx + d$). Visually, a linear polynomial represents a straight line, while a quadratic polynomial represents a U-shaped curve known as a parabola.

A 'Zero of a Polynomial' $p(x)$ is a number $c$ such that $p(c) = 0$. Visually, if you plot the polynomial $y = p(x)$ on a graph, the zeroes are the $x$-coordinates of the specific points where the graph crosses or touches the horizontal $x$-axis.