1. The graphs of are given in Fig. 2.10 below, for some polynomials . Find the number of zeroes of , in each case.

Question

Homi · Accepted Answer

Understand the Question

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 $y = 0$ ).
Therefore, the number of zeroes of $p(x)$ is simply the number of times the graph intersects or touches the $x$ -axis.

(i) Find the number of zeroes of $p(x)$ for graph (i).

Step 1 · Analyze Graph (i)

The graph is a straight line parallel to the $x$ -axis and does not intersect or touch the $x$ -axis at any point. Diagram 1

$\text{Number of zeroes} = 0$

Answer

(i) $0$

(ii) Find the number of zeroes of $p(x)$ for graph (ii).

Step 1 · Analyze Graph (ii)

The graph intersects the $x$ -axis at $3$ distinct points. $\text{Number of zeroes} = 3$

Answer

(ii) $3$

(iii) Find the number of zeroes of $p(x)$ for graph (iii).

Step 1 · Analyze Graph (iii)

The graph intersects the $x$ -axis at only $1$ point. $\text{Number of zeroes} = 1$

Answer

(iii) $1$

(iv) Find the number of zeroes of $p(x)$ for graph (iv).

Step 1 · Analyze Graph (iv)

The graph is a parabola that intersects the $x$ -axis at $2$ distinct points. $\text{Number of zeroes} = 2$

Answer

(iv) $2$

(v) Find the number of zeroes of $p(x)$ for graph (v).

Step 1 · Analyze Graph (v)

The graph intersects the $x$ -axis at $4$ distinct points. $\text{Number of zeroes} = 4$

Answer

(v) $4$

(vi) Find the number of zeroes of $p(x)$ for graph (vi).

Step 1 · Analyze Graph (vi)

The graph touches/intersects the $x$ -axis at $3$ distinct points. $\text{Number of zeroes} = 3$

Answer

(vi) $3$

Common Mistakes

Counting $y$ -axis Intercepts: Mistakenly counting points where the graph crosses the $y$ -axis. Zeroes of $y = p(x)$ correspond strictly to intersections with the $x$ -axis ( $y = 0$ ).
Ignoring Points of Contact: Forgetting that a point where the graph just touches the $x$ -axis (without crossing) still counts as a zero.

Question 1