Question

The graph of y=p(x) is given in the figure, for some polynomials p(x). Find the number of zeros of p(x), in each case.

Answer 1

Hint: The number of zeros of a polynomial is determined by the number of times the given curve crosses the x-axis.
The zeros of a polynomial are the solutions of the equation p(x)=0, where p(x) represents the polynomial. If a given polynomial is represented on the graph, we get a curve which may or may not cross x-axis or the y-axis, but in case of zeros, the curve passes through the x-axis.
In the given question, the graph is given for y=p(x); hence we have to observe the graph to find the number of zeros.


Complete step by step solution:
The number of zeros of the polynomial is determined by checking the number of times the curve crossed the x-axis,

In this graph polynomial is a straight line passing through the y-axis, but the graph does not intercept x-axis at any point; hence the number of zeros for the polynomial is zero in p(x).

In this graph, the curve crosses through the x-axis one time; hence the number of zeros for the polynomial is one in p(x).

In this graph, the curve crosses through the x-axis three times; hence the number of zeros for the polynomial is three in p(x).

In this graph, the curve crosses through the x-axis two times; hence the number of zeros for the polynomial is two in p(x).

In this graph, the curve crosses through the x-axis four times; hence the number of zeros for the polynomial is four in p(x).

In this graph, the curve crosses through the x-axis one time and touches the x-axis two times; hence the number of zeros for the polynomial is three in p(x).

Note: XY graphs are used to study the nature of the polynomial equation, if a polynomial is represented in an x-y graph, then the graph shows the nature of the curve.