Polynomials - Geometrical Meaning of the Zeroes of a Polynomial

Definition of a Zero: A real number $k$ is said to be a zero of a polynomial $p(x)$ if $p(k) = 0$. Geometrically, the zeroes of a polynomial $p(x)$ are the x-coordinates of the points where the graph of $y = p(x)$ intersects the x-axis.

Linear Polynomials: For a linear polynomial $ax + b$ where $a \neq 0$, the graph is a straight line. This line intersects the x-axis at exactly one point, $(-\frac{b}{a}, 0)$. Therefore, every linear polynomial has exactly one zero.

Quadratic Polynomials and Parabolas: The graph of a quadratic polynomial $p(x) = ax^2 + bx + c$ is a curve called a parabola. If $a > 0$, the parabola opens upwards (like a 'U'), and if $a < 0$, the parabola opens downwards (like an inverted 'U').

Number of Zeroes for Quadratics: A quadratic polynomial can have at most 2 zeroes. Visually, there are three cases: the parabola can intersect the x-axis at two distinct points (2 zeroes), touch the x-axis at exactly one point (1 zero/two equal zeroes), or not touch the x-axis at all (no real zeroes).

Cubic Polynomials: A cubic polynomial $p(x) = ax^3 + bx^2 + cx + d$ can have at most 3 zeroes. Geometrically, the curve of a cubic polynomial can cross the x-axis at 1, 2, or 3 points. Unlike quadratics, a cubic polynomial always has at least one real zero because the curve must cross the x-axis as it extends from negative infinity to positive infinity.

General Rule for Degree $n$: In general, a polynomial $p(x)$ of degree $n$ has at most $n$ zeroes. This means the graph of $y = p(x)$ can intersect the x-axis at a maximum of $n$ points.

Interpreting Graphs: To find the number of zeroes from a given graph of $y = p(x)$, simply count how many times the curve touches or crosses the horizontal x-axis. Points where the graph crosses the vertical y-axis are ignored as they represent the value $p(0)$, not the zeroes of the polynomial.