02 Class-X - Polynomials (AP, TG and CBSE)

2.1 Introduction

2.2 Geometrical Meaning of the Zeroes of a Polynomial

Example 1-i: Look at the graph in Fig.  given below. It is the graph of y = p(x), where p(x) is a polynomial. For  the graph, find the number of zeroes of p(x). 

Example 1-ii : Look at the graph in Fig.  given below. It is the graph of y = p(x), where p(x) is a polynomial. For  the graph, find the number of zeroes of p(x). 

Example 1-iii : Look at the graph in Fig.  given below. It is the graph of y = p(x), where p(x) is a polynomial. For  the graph, find the number of zeroes of p(x).  

Example 1-iv : Look at the graph in Fig.  given below. It is the graph of y = p(x), where p(x) is a polynomial. For  the graph, find the number of zeroes of p(x).  

Example 1-v : Look at the graph in Fig.  given below. It is the graph of y = p(x), where p(x) is a polynomial. For  the graph, find the number of zeroes of p(x). 

Example 1-vi : Look at the graph in Fig.  given below. It is the graph of y = p(x), where p(x) is a polynomial. For  the graph, find the number of zeroes of p(x).   

EXERCISE 2.1 

1(i). The graphs of y = p(x) are given in Fig.  below, for some polynomial p(x). Find the number of zeroes of p(x)

1(ii). The graphs of y = p(x) are given in Fig.  below, for some polynomial p(x). Find the number of zeroes of p(x)

 

1(iii).The graphs of y = p(x) are given in Fig.  below, for some polynomial p(x). Find the number of zeroes of p(x)

 

1(iv).The graphs of y = p(x) are given in Fig.  below, for some polynomial p(x). Find the number of zeroes of p(x)

 

1(v).The graphs of y = p(x) are given in Fig.  below, for some polynomial p(x). Find the number of zeroes of p(x)

 

1(vi).The graphs of y = p(x) are given in Fig.  below, for some polynomial p(x). Find the number of zeroes of p(x) 

2.3 Relationship between Zeroes and Coefficients of a Polynomial

 Example 2 : Find the zeroes of the quadratic polynomial x2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.

Example 3 : Find the zeroes of the polynomial x²  – 3 and verify the relationship between the zeroes and the coefficients

Example 4 : Find a quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively.

Example 5* : Verify that 3, –1, 1/3  are the zeroes of the cubic polynomial p(x) = 3x³  – 5x ² – 11x – 3, and then verify the relationship between the zeroes and the coefficients  

* Not from the examination point of view.

 EXERCISE 2.2 

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. 

(i) x²  – 2x – 8 

(ii) 4s²  – 4s + 1 

(iii) 6x²  – 3 – 7x

 (iv) 4u²  + 8u 

(v) t²  – 15 

(vi) 3x² – x – 4 

2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. 

(i) 1//4 , -1 

(ii) √2 , 1/3 

(iii) 0, √5

 (iv) 1, 1 

(v) 1/4 ,  -1/4  

(vi) 4, 1

2.4 Summary 

1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively. 

2. A quadratic polynomial in x with real coefficients is of the form ax2 + bx + c, where a, b, c are real numbers with a ≠ 0. 

3. The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x) intersects the x - axis. 

4. A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes. 

5. If 𝛼 and 𝛽 are the zeroes of the quadratic polynomial ax²  + bx + c, then  𝛼+𝛽 = -b/a,  𝛼𝛽 = c/a 

6. If 𝛼,𝛽, 𝛾are the zeroes of the cubic polynomial ax³  + bx² + cx + d, then   𝛼+𝛽+ 𝛾 = -b/a    𝛼𝛽+ +𝛽𝛾+𝛾𝛼 = c/a, and  𝛼+𝛽+ 𝛾 = -d/a

Objective Questions (MCQs)

1. If x = 3 is one of the roots of the quadratic equation x² – 2kx – 6 = 0, then the value of k is

1. 1/2
2. 1/2
3. 3
4. 2   

2.The quadratic polynomial whose sum of zeroes is -3 and product of zeroes is  [AP SSC ARCH 2025 ]

1. x² + 3x + 2  

2.  x² – 3x + 2 

3. x² – 3x – 2

4. x² + 3x – 2 

3. Assertion : The polynomial x³ -2x²-7x-12 is a cubic polynomial.

Reason : Because the degree of the polynomial is 3.

which of the following is True? [ AP MP 2025 ]

A) Both Assertion and Reason are true, Reason is supporting the assertion.

B) Both Assertion and Reason are true, but Reason is not supporting the assertion.

C) Assertion is true but the reason is false.

D) Assertion  is false, but the reason is true.

4.The    zero of the polynomial ax+b is  [ AP MP 2025 ]

1] b/a  B] a/b   C]  -a/b   D]  -b/a

5.The    zero of the polynomial 2x-3 is 

1]⅔  B]3/2  C] -⅔  D] -3/2

Miscellaneous

1.Writ an example for nominal having degree 6.  [AP SSC MARCH 2025 ]

2.Find the zeroes of the quadratic polynomial x² + 7x + 10 [ AP SSC MARCH 2025 ]

3..Find the zeroes of the quadratic polynomial 4s² – 4s + 1 and verify the relationship between the zeroes and the coefficients.

4.Due to heavy storm an electric wire got bent as shown in the figure. It followed a mathematical shape. Answer the following questions below:  [AP SPQ 2025 ]

     a)Name the shape in which the wire is bent.

     b)How many zeros are there for the polynomial (Shape of the wire )

     c)The zeroes of the polynomial are...

     d)Sum of the zeroes of the polynomial.

 5..Observe the following graph and answer the following questions. [AP SSC M 2025]

+

 

     a)Name the shape of the graph

     b)How  many zeroes of the polynomial are there in the graph?

     c)What are the zeroes of the polynomial in the graph?

    d)Write the sum of zeroes.

6.Observe the graph and answer the following questions.

i) What is the shape of the graph

ii) How many zeroes it has

iii) What are the zeroes? 

iv)How many times graph intersect x-axis

7.Find the zeros of the quadratic polynomial x²+7x+10 [AP SSC M 2025 ]

8.Find a quadratic polynomial whose sum and product of its zeroes are -1 and ¼ respectively.