Question 1
The graph of a polynomial $y = p(x)$ is given. Find the number of zeroes of the polynomial if the graph intersects the $x$-axis at exactly one point.
Geometrical Meaning of Zeroes
The zeroes of a polynomial $y = p(x)$ are the $x$-coordinates of the points where the graph intersects or touches the $x$-axis (i.e. where $y = 0$).
Analysis of Graph
Since the curve intersects the $x$-axis at exactly one point, there is only one real value of $x$ for which $p(x) = 0$.
$$\therefore \text{Number of zeroes} = 1$$
Question 2
A graph of a polynomial $y = p(x)$ intersects the $x$-axis at two distinct points. Find the number of zeroes of $p(x)$.
Points of Intersection
The number of real zeroes is equal to the number of distinct points where the curve cuts the $x$-axis.
Here, the curve intersects the $x$-axis at two distinct points.
Here, the curve intersects the $x$-axis at two distinct points.
$$\therefore \text{Number of zeroes} = 2$$
Question 3
The graph of a polynomial $y = p(x)$ touches the $x$-axis at exactly one point (tangent to $x$-axis). Find the number of zeroes.
Tangent Condition
When a polynomial curve only touches the $x$-axis without crossing it, the two intersection points coincide (representing equal real roots).
$$\therefore \text{Number of zeroes} = 1 \quad \text{(coincident roots)}$$
Question 4
The graph of a polynomial $y = p(x)$ is a line parallel to the $x$-axis (does not intersect the $x$-axis at any point). Find the number of zeroes.
Zero Intersections
Since the graph does not meet or intersect the $x$-axis anywhere, there is no real value of $x$ for which $p(x) = 0$.
$$\therefore \text{Number of zeroes} = 0$$
Question 5
A polynomial graph $y = p(x)$ cuts the $x$-axis at three distinct points. How many zeroes does the polynomial have?
Three Intersections
Each point of intersection with the $x$-axis corresponds to a distinct real zero of the polynomial $p(x)$.
Since the graph cuts the $x$-axis at three distinct points, it has three real zeroes.
Since the graph cuts the $x$-axis at three distinct points, it has three real zeroes.
$$\therefore \text{Number of zeroes} = 3$$