Geometrical Meaning of the Zeroes of a Polynomial

The zeroes of a polynomial are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.

A quadratic polynomial can have only:

Case (i): Two zeroes

• Graph cuts the x-axis at two distinct points

• Two distinct real zeroes

Case (ii): One zero

• Graph touches the x-axis at exactly one point

• One real (repeated) zero

Case (iii): No zero

• The graph does not touch the x-axis at all

• No real zero

A polynomial of degree n can have at most n zeroes.