Exercise 2.1, 1

The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

Solution: 

Understanding Zeroes of a Polynomial

The zeroes of a polynomial are the x-coordinates where the graph intersects the x-axis.

Now, let’s check each graph:

1. Graph (i):

   • The graph does not intersect the x-axis.
   • Number of zeroes = 0.

2. Graph (ii):

   • The graph intersects the x-axis at one point.
   • Number of zeroes = 1.

3. Graph (iii):

   • The graph intersects the x-axis at two points.
   • Number of zeroes = 2.

4. Graph (iv):

   • The graph intersects the x-axis at one point, but it touches it instead of crossing.
   • Number of zeroes = 1 (double root).

5. Graph (v):

   • The graph intersects the x-axis at three points.
   • Number of zeroes = 3.

6. Graph (vi):

   • The graph intersects the x-axis at four points.
   • Number of zeroes = 4.