+ bx + c , the following three cases can happen: Fig. . Case (i) : Here, the graph cuts x -axis at two distinct points A and A . The x -coordinates of A and A are the two zeroes of the quadratic polynomial ax + bx + c in this case (see Fig.
Case (ii) : Here, the graph cuts the x -axis at exactly one point, i.e., at two coincident points. So, the two points A and A of Case (i) coincide here to become one point A (see Fig. . ).
Fig. . The x -coordinate of A is the only zero for the quadratic polynomial ax + bx + c in this case. Case (iii) : Here, the graph is either completely above the x -axis or completely below the x -axis.
So, it does not cut the x -axis at any point (see Fig. . ). Fig.
. So, the quadratic polynomial ax + bx + c has no zero in this case. So, you can see geometrically that a quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero. This also means that a polynomial of degree has atmost two zeroes.
Now, what do you expect the geometrical meaning of the zeroes of a cubic polynomial to be? Let us find out. Consider the cubic polynomial x – x . To see what the graph of y = x – x looks like, let us list a few values of y corresponding to a few values for x as shown in Table .
. Table . – – y = x – x – Locating the points of the table on a graph paper and drawing the graph, we see that the graph of y = x – x actually looks like the one given in Fig. .
. We see from the table above that – , and are zeroes of the cubic polynomial x