Graph of y = p(x) and the X-axis
For any polynomial y = p(x), the zeroes are the x-coordinates of the points where the graph intersects or touches the X-axis (i.e. where y = 0).
A polynomial p(x) of degree n intersects the X-axis at at most n points. Therefore, a polynomial of degree n has at most n zeroes.
Graphs of Linear & Quadratic Polynomials
Graph of Quadratic Polynomial y = ax² + bx + c (Parabola)
The graph of a quadratic polynomial is a curve called a parabola. It can open upwards (a > 0) or downwards (a < 0):
Here, the parabola cuts the X-axis at two points (α, 0) and (β, 0). Thus, the polynomial has 2 distinct real zeroes: x = α and x = β.
3 Possible Cases for Quadratic Graphs
Cuts X-axis at 2 distinct points
The polynomial has 2 distinct zeroes.Touches X-axis at exactly 1 point
The 2 points coincide, giving 2 equal zeroes (1 unique zero).Remains completely above or below X-axis
Does not intersect X-axis, so it has 0 real zeroes.Key Takeaways
Number of zeroes = Number of points where graph y = p(x) cuts/touches the X-axis.
Graph of a linear polynomial is a straight line (at most 1 zero).
Graph of a quadratic polynomial is a parabola (at most 2 zeroes).