Theory:

A polynomial of the form \(ax^3 + bx^2 + cx + d\), \(a \neq 0\) is a cubic polynomial.
 
Consider the graph of \(y = 2x^3 - 3x\).
 
\(x\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
\(2x^3\)
\(2(-2)^3\)
 
\(=\) \(2(-8)\)
 
\(=\) \(-16\)
\(2(-1)^3\)
 
\(=\) \(2(-1)\)
 
\(=\) \(-2\)
\(2(0)^3\)
 
\(=\) \(2(0)\)
 
\(=\) \(0\)
\(2(1)^3\)
 
\(=\) \(2(1)\)
 
\(=\) \(2\)
\(2(2)^3\)
 
\(=\) \(2(8)\)
 
\(=\) \(16\)
\(3x\)
\(3 \times -2\)
 
\(=\) \(-6\)
\(3 \times -1\)
 
\(=\) \(-3\)
\(3 \times 0\)
 
\(=\) \(0\)
\(3 \times 1\)
 
\(=\) \(3\)
\(3 \times 2\)
 
\(=\) \(6\)
\(2x^2 - 3x\)
\(-16\) \(+\) \(6\)
 
\(=\) \(-10\)
\(-2\) \(+\) \(3\)
 
\(=\) \(1\)
\(0\) \(-\) \(0\)
 
\(=\) \(0\)
\(2\) \(-\) \(3\)
 
\(=\) \(-1\)
\(16\) \(-\) \(6\)
 
\(=\) \(10\)
\(y\) \(=\) \(2x^2 - 3x\)
\(-10\)
\(1\)
\(0\)
\(-1\)
\(10\)
 
Join the coordinates \((-2, -10)\), \((-1, 1)\), \((0, 0)\), \((1, -1)\) and \((2, 10)\) by a smooth curve so as to obtain the graph of \(y = 2x^2 - 3x\).
 
cubic.png
 
It is observed that, the graph of the polynomial \(y = 2x^2 - 3x\) intersects the \(x\) \(-\) axis at three points.
 
Thus, we can say that the zero of a cubic polynomial is the \(x\) \(-\) coordinates of the point where the graph of the polynomial intersects the \(x\) \(-\) axis.
 
Let us discuss few cases of graphs of a cubic polynomial.
 
Case 1: The graph of the cubic polynomial cuts the \(x\) \(-\) axis at one point, as shown below.
 
C1.png
 
In this case, the number of zeroes of the polynomial is \(1\).
 
Case 2: The graph of the cubic polynomial cuts the \(x\) \(-\) axis at two points, as shown below.
 
C2.png
 
In this case, the number of zeroes of the polynomial is \(2\).
 
Case 3: The graph of the cubic polynomial cuts the \(x\) \(-\) axis at three points, as shown below.
 
C3.png
 
In this case, the number of zeroes of the polynomial is \(3\).
The cubic polynomial \(ax^3 + bx^2 + cx + d\), \(a \neq 0\), has at most three zeroes which is the \(x\) \(-\) coordinates of the point where the graph of the polynomial intersects the \(x\) \(-\) axis.
Important!
In general, the graph of the polynomial of \(n\) degree intersects the \(x\) \(-\) axis at atmost \(n\) points.
 
In other words, the polynomial of \(n\) degree has atmost \(n\) zeroes.