How to find zeroes of polynomials in graph class 10?

How to find zeroes of polynomials in polynomial graphs?

It is an important topic in class 10th students who opt for mathematics. Now, the CBSE board asks questions in MCQ types. So, it is an important topic for the MCQ level. I provide important questions and previous years' questions. It helps to understand the intent of the board questions.

In this article, I will tell you easy tricks to find zeroes using types & nature curves in graphs. First, you have to understand the types of polynomials based on Degree. Nature of curves in polynomial graphs based on the Degree of polynomials.

 

Below, I have provided six graphs of quadratic polynomial conditions. There are six conditions of the curve. It helps you to understand the nature of Parabola in graphs. After analysing these graphs, you can understand the number of zeroes. There are different values of the coefficient of x-term in quadratic polynomials which helps to understand the nature of the curve. Relation of zeroes and coefficients

Types of Polynomials based on Degree of Polynomial.
Degree of polynomials	Examples	Types of Polynomial
Not Defined	0	Zero
0	a,a≠0	Constant (non-zero)
1	ax+b,a≠0	Linear Polynomial
2	ax2+bx+c,a≠0	Quadratic Polynomial
3	ax3+bx2+cx+d,a≠0	Cubic Polynomial
4	ax4+bx3+cx2+d,a≠0	Quatric Polynomial
5	ax5+bx4+cx3+dx2+e,a≠0	Quintic Polynomial

It is an easy trick to find several zeroes after analysing polynomial graphs. I hope it helps to improve your analysing ability of polynomial graphs. it is a short-cut method to find the number of the zeroes of a polynomial.

Nature of Curve based on Degree of Polynomial
Degree Of Polynomial	Types of Polynomial	Nature of Curve	Number of Zeroes
1	Linear Polynomial	straight line	One
2	Quadratic Polynomial	Parabola	Two
3	Cubic Polynomial	S-shaped	Three
4	Quatric Polynomial	W-shaped	Four

{Graph of a Linear Polynomial } 

Let's consider a linear polynomial f(x) = ax+b, a≠0. In this graph, y = ax+b is a straight line. That is why f(x) = ax+b is a linear polynomial. since two points determine a straight line, These points need to be plotted to draw the line y = ax+b. The line represented by y = ax+b across the x-axis at exactly one point

{Graph of a Quadratic Polynomial }

Let's consider a Quadratic Polynomial f(x) = ax²+bx+c where a≠0. In this graph y = ax²+bx+c is a parabola. Parabola y = ax²+bx+c cuts or touches at two points of the axis in this graph. These two points are two zeroes of quadratic polynomials. The parabola may open upward or downward in graphs. This curve(Parabola) is U-shaped. The nature of curves depends on a, b & c(coefficient) in quadratic polynomials.  All cases are provided above.

Case 1: When polynomial ax²+bx+c is factor into distinct linear factors:

In this case: the graph of ax²+bx+c or the curve y = ax²+bx+c cuts the X-axis at two distinct points. The x-coordinates of these  points are the two zeros  of the  polynomial

Case 2 When polynomial ax²+bx+c is factorizable into two equal factors. 

In this case: the graph of the polynomial ax+bx+c or the curve y = ax²+bx+c touches the X-axis at point (-b/2a,0). the x-coordinates of this point give two equal zeros to the polynomial.

Case 3: When polynomial ax²+bx+c is not a factorizable.

In this case: the graph of the polynomial ax²+bx+c or the curve y = ax²+bx+c does not cut or touch the x-axis. Parabola y = ax+bx+c opens upwards and remains completely above the x-axis if a>0. the parabola opens downward and remains completely below the x-axis if a<0.

{Graph of a Cubic Polynomial}

Let's consider a Cubic Polynomial f(x) = ax³+bx²+cx+d where a≠0. In this graph, y = ax³+bx²+cx+d represents an S-shaped curve. This S-shaped curve cuts or touches at three points at the axis in this graph. These points are zeroes of the cubic polynomial. The curve in a cubic polynomial graph depends on the value of a, b, c & d (coefficient). In some cubic polynomials, it does not represent an S-shaped curve.

{Graph of a Quartic Polynomial}

Let's consider a Quartic Polynomial f(x) = ax⁴+bx³+cx²+dx+e where a≠0. In this graph y = ax⁴+bx³+cx²+dx+e represent a W-shaped curve. This W-shaped curve cuts or touches at four points at the axis in this graph. These points are zeroes of Quartic Polynomials. The W-shaped curve graph depends upon the value of a,b,c,d & e (coefficient).

Some Important Previous Years Questions based on Quadratic Polynomial. These Questions were asked in the Board examination by the CBSE Board. 2024-25

Q1. In the given figure, the graph of a polynomial p(x) is given. Number of zeroes  of p(x) is : 

[CBSE 2023C]

Q2. In the given figure, the graph of a polynomial p(x) is given below. Find the number of zeroes  of p(x) is : 

[CBSE 2023C]

Q3. The graph of y = p(x) is given in the adjoining figure. Zeroes of the polynomial p(x) are:

[CBSE 2023]

Q4. The graph of y = p(x) is given, for a polynomial p(x). The number of zeroes of p(x) from the graph is.

[CBSE 2023]

>Q5. The graph of y = p(x) is shown in the figure for some polynomial p(x). The number of zeroes of p(x) is/are:

[CBSE 2023]

Q6. The graph of y = f(x) is shown in the figure for some polynomial f(x).

[CBSE 2023]