– x . Observe that – , and are, in fact, the x -coordinates of the only points where the graph of y = x – x intersects the x -axis. Since the curve meets the x -axis in only these points, their x -coordinates are the only zeroes of the polynomial. Let us take a few more examples.
Consider the cubic polynomials x and x – x . We draw the graphs of y = x and y = x – x in Fig. . and Fig.
. respectively. Fig. .
Note that is the only zero of the polynomial x . Also, from Fig. . , you can see that is the x -coordinate of the only point where the graph of y = x intersects the x -axis.
Similarly, since x – x = x ( x – ), and are the only zeroes of the polynomial x – x . Also, from Fig. . , these values are the x -coordinates of the only points where the graph of y = x – x intersects the x -axis.
From the examples above, we see that there are at most zeroes for any cubic polynomial. In other words, any polynomial of degree can have at most three zeroes. Remark : In general, given a polynomial p ( x ) of degree n , the graph of y = p ( x ) intersects the x -axis at atmost n points. Therefore, a polynomial p ( x ) of degree n has at most n zeroes.
Example : Look at the graphs in Fig. . given below. Each is the graph of y = p ( x ), where p ( x ) is a polynomial.
For each of the graphs, find the number of zeroes of p ( x ). Fig. . Solution : (i) The number of zeroes is as the graph intersects