Polynomials - Zeroes of a Polynomial

A zero of a polynomial $p(x)$ is a real number $c$ such that $p(c) = 0$. Visually, if we think of the polynomial as a machine where we drop in a value $x$, the 'zeroes' are the specific inputs that result in an output of exactly $0$.

A non-zero constant polynomial, such as $p(x) = 5$, has no zeroes because the value remains constant and never reaches zero. Visually, this can be imagined as a horizontal line that never touches or crosses the $x$-axis.

The zero polynomial, denoted by $p(x) = 0$, is unique because every real number is considered a zero for it. In a graphical sense, the zero polynomial lies exactly on the $x$-axis at all points.

Every linear polynomial in one variable has one and only one zero. For a linear equation $ax + b = 0$, the zero represents the unique point where the straight line graph of the polynomial intersects the $x$-axis.

A polynomial can have more than one zero, but the total number of zeroes cannot exceed its degree. For example, a quadratic polynomial of degree $2$ can be visualized as a U-shaped curve (parabola) that can cross the $x$-axis at most at two distinct points.

It is important to note that a zero of a polynomial does not have to be the number $0$. A polynomial like $p(x) = x - 1$ has a zero at $1$. Conversely, $0$ can be a zero of a polynomial, as seen in $p(x) = x^2$, where the graph touches the origin $(0,0)$.

To find the zero of any polynomial $p(x)$, we solve the polynomial equation $p(x) = 0$. This algebraic process is the method used to identify the 'roots' of the equation, which are identical to the 'zeroes' of the polynomial.